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\title{Controllability of a fluid-structure interaction system coupling the Navier--Stokes system and a damped beam equation}

\author{\firstname{R\'emi} \lastname{Buffe}}
\address{Universit\'e de Lorraine, CNRS, Inria, IECL, 54000 Nancy, France}
\email{remi.buffe@univ-lorraine.fr}

\author{\firstname{Tak\'eo} \lastname{Takahashi}\IsCorresp}
\address[1]{Universit\'e de Lorraine, CNRS, Inria, IECL, 54000 Nancy, France}
\email{takeo.takahashi@inria.fr}

\thanks{The second author was partially supported by the Agence Nationale de la Recherche, Project TRECOS, ANR-20-CE40-0009}
\CDRGrant[ANR]{ANR-20-CE40-0009}

\begin{abstract}
We show the local null-controllability of a fluid-structure interaction system coupling a viscous incompressible fluid with a damped beam located on a part of its boundary. The controls act on arbitrary small parts of the fluid domain and of the beam domain. In order to show the result, we first use a change of variables and a linearization to reduce the problem to the null-controllability of a Stokes-beam system in a cylindrical domain. We obtain this property by combining Carleman inequalities for the heat equation, for the damped beam equation and for the Laplace equation with high-frequency estimates. Then, the result on the nonlinear system is obtained by a fixed-point argument.
\end{abstract}

\keywords{\kwd{Null controllability}
\kwd{Navier--Stokes systems}
\kwd{Carleman estimates}
\kwd{fluid-structure interaction systems}}

\subjclass{76D05, 35Q30, 74F10, 76D55, 76D27, 93B05, 93B07, 93C10}

\begin{document}
\maketitle
%\tableofcontents

\section{Introduction}
We consider a fluid-structure interaction system composed by a viscous incompressible fluid, modeled by the Navier--Stokes system, and by an elastic structure located at a part of the boundary of the fluid domain. We assume that the structure displacement is governed by a damped beam equation. The corresponding model has been introduced in~\cite{zbMATH01502146} as a first model to study the blood flow in vessels. To simplify our work, we consider here a particular geometry in dimension~2 of space (see Figure~\ref{F1}). The fluid domain is confined into an infinite strip where the bottom boundary is fixed and where the top boundary corresponds to the beam. We also assume periodic condition in the $x_1$ variables. To be more precise, we set
\[
\mathcal{I}:=\mathbb{R}/(2\pi\mathbb{Z}),
\]
and for any deformation $\zeta : \mathcal{I} \to (-1,\infty)$, we consider the fluid domain associated with this deformation:
\begin{equation}\label{gev0.0}
\Omega_{\zeta}= \left \{(x_1,x_2)\in \mathcal{I}\times \mathbb{R} \ ; \ x_2\in (0,1+\zeta(x_1))\right \}.
\end{equation}
Then the fluid-structure interaction system writes
\begin{equation}\label{tak2.3}
\begin{cases}
\partial_t w +(w\cdot \nabla) w-\div \mathbb{T}(w,\pi) = 1_{\omega} f & t>0, \ x\in \Omega_{\zeta(t)},\\
\div w = 0 & t>0, \ x\in \Omega_{\zeta(t)},\\
w(t,x_1,1+\zeta(t,x_1)) = (\partial_t \zeta)(t,x_1) e_2, & t>0, \ x_1\in \mathcal{I},\\
w = 0 & t>0, \ x\in \Gamma_{0},\\
\partial_{tt} \zeta + \alpha_1 \partial_{x_1}^4 \zeta-\alpha_2 \partial_{x_1}^2 \zeta -\alpha_3 \partial_{t}\partial_{x_1}^2 \zeta =-\widetilde{\mathbb{H}}_{\zeta}(w,\pi) +1_{\mathcal{J}} g& t>0, \ x_1\in \mathcal{I},\\
w(0,\cdot\,)=w^{0} \quad \text{in} \ \Omega_{\zeta_1^0},\quad \zeta(0,\cdot\,)=\zeta_1^0, & \partial_t\zeta (0,\cdot\,)=\zeta_2^0 \quad \text{in} \ \mathcal{I},
\end{cases}
\end{equation}
where
\[
\alpha_1>0, \quad \alpha_2\geq 0, \quad \alpha_3>0,
\]
and where
\[
\Gamma_0=\mathcal{I}\times \{0\}.
\]
In the above system, we have used the following notations: $(e_1, e_2)$ is the canonical basis of $\mathbb{R}^2$ and
\begin{gather}\label{ws4.6}
\mathbb{T}(w,\pi) = 2 \mathbb{D}(w)- \pi I_2,\quad \mathbb{D}(w)= \frac{1}{2}\left(\nabla w + (\nabla w)^*\right),
\\ \label{ws4.7}
\widetilde{\mathbb{H}}_{\zeta}(w,\pi)(t,x_1)= \left[(1+|\partial_{x_1} \zeta|^2)^{1/2} \left[\mathbb{T}(w,\pi)n\right](t,x_1,1+\zeta(t,x_1))\cdot e_2 \right].
\end{gather}
We have also denoted by $n$ the unit exterior normal to $\Omega_{\zeta(t)}$. In~\eqref{tak2.3}, $w$ and $\pi$ are respectively the velocity and the pressure of the fluid and they satisfy the Navier--Stokes system (two first lines), with no-slip boundary conditions (third and forth equations). The elastic displacement satisfies the damped beam equation written in the fifth line of~\eqref{tak2.3}. Finally, our aim is to control~\eqref{tak2.3} by using two distributed controls $f$ and $g$ respectively localized in an arbitrary small nonempty open subset $\omega$ of $\Omega$ and in an arbitrary small nonempty open subset $\mathcal{J}$ of $\mathcal{I}$.

Let us remark that the well-posedness and the stabilization of system~\eqref{tak2.3} have been already studied in the literature. Let us quote some of the corresponding articles:~\cite{ChaDesEst2005a} (existence of weak solutions), \cite{Bei2004a, Leq2011a, MR3466847, Debayan} (existence of strong solutions), \cite{MR2745779} (stabilization of strong solutions), \cite{MR3619065} (stabilization of weak solutions around a stationary state). We can also mention some works devoted to the case $\delta=0$ (undamped beam equation/wave equation):~\cite{Gra2008a, MuhaCanic, CanicMuhaBukac} (weak solutions), \cite{grandmont:hal-01567661, plat, nonplat, wave} (strong solutions). Some authors have tackled the study of more complex models:~\cite{LR14, Leng14} (linear elastic Koiter shell), \cite{MC-arma} (dynamic pressure boundary conditions), \cite{Boris-C-13, Boris-C-15} (3D cylindrical domain with nonlinear elastic cylindrical Koiter shell), \cite{MR4042350} and~\cite{MR4192391} (nonlinear elastic and thermoelastic plate equations), \cite{MR4253566, MR4183914} (compressible fluids), etc.

A standard strategy to study this kind of systems consists in using a change of variables to write the fluid system into a cylindrical domain, and then in linearizing the system after this transformation. A large part of the work is thus devoted to the corresponding linear system, the results for the nonlinear system are deduced by estimating the coefficients coming from the change of variables and by using a fixed-point argument. We follow here this approach and after a change of variable and a linearization (see \cref{sec_proof} for the details), we are reduced to work on the spatial domain
\[
\Omega:=\Omega_0=\mathcal{I}\times (0,1)
\]
(see Figure~\ref{F1}) and to show the null controllability of the following linear system
\begin{equation}\label{ns0.1}
\begin{cases}
\partial_{t} w - \Delta w + \nabla \pi = 1_{\omega} f & \text{in} \ (0,T)\times \Omega,\\
\div w=0& \text{in} \ (0,T)\times \Omega,\\
w=0 & \text{on} \ (0,T)\times \Gamma_0,\\
w=(\partial_t \zeta) e_2 & \text{on} \ (0,T)\times \Gamma_1,\\
\partial_{t}^2 \zeta + \alpha_1 \partial_{x_1}^4 \zeta-\alpha_2 \partial_{x_1}^2 \zeta -\alpha_3 \partial_{t}\partial_{x_1}^2 \zeta = -\mathbb{T}(w,\pi)n\cdot e_2 +1_{\mathcal{J}} g& \text{in}\ (0,T)\times \mathcal{I},\\
w(0,\cdot\,)=w^{0} & \text{in} \ \Omega,\\
\zeta(0,\cdot\,)=\zeta_1^0,\quad \partial_t\zeta (0,\cdot\,)=\zeta_2^0 & \text{in} \ \mathcal{I},
\end{cases}
\end{equation}
where
\[
\Gamma_1=\mathcal{I}\times \{1\}.
\]
In what follows, to simplify the notation, we take
\[
\alpha_1=\alpha_2=\alpha_3=1.
\]
The values of these constants do not play any role in our study. As it is standard (see, for instance, \cite[Theorem~11.2.1, p.~357]{TucsnakWeiss}), the controllability of~\eqref{ns0.1} is equivalent to an observability inequality for the adjoint system

\begin{figure}\label{F1}
\begin{center}
\begin{tikzpicture}
\begin{scope}[xshift=200]
\draw (0,0) -- (5,0);
\draw (0,2) -- (0,0);
\draw (5,0) -- (5,2);
\draw[dashed] (0,2) -- (5,2);
\draw (2.5,1) node {$\Omega_\zeta$};
\draw (2.5,0) node[below] {$\Gamma_{0}$};
\draw (0,1) node {$-$};
\draw (0,0.95) node {$-$};
\draw (5,1) node {$-$};
\draw (5,0.95) node {$-$};
\draw (2.5,2.3) node[above] {$\{x_2=1+\zeta(x_1)\}$};
\draw [line width=0.4mm, blue, domain=0:5,samples=50] plot (\x, {2-0.04*sin(2*pi*\x r)+0.2*sin(pi*\x r)});
\draw (0,0) node[below] {$0$};
\draw (5,0) node[below] {$2\pi$};
\draw (0,2) node[left] {$1$};
\end{scope}
\draw (0,0) -- (5,0);
\draw (0,2) -- (0,0);
\draw (5,0) -- (5,2);
\draw[line width=0.4mm, blue] (0,2) -- (5,2);
\draw (2.5,1) node {$\Omega$};
\draw (2.5,0) node[below] {$\Gamma_{0}$};
\draw (0,1) node {$-$};
\draw (0,0.95) node {$-$};
\draw (5,1) node {$-$};
\draw (5,0.95) node {$-$};
\draw (2.5,2) node[above] {$\Gamma_{1}$ {\color{blue}(beam)}};
\draw (0,0) node[below] {$0$};
\draw (5,0) node[below] {$2\pi$};
\draw (0,2) node[left] {$1$};
\end{tikzpicture}
\end{center}
\caption{Our geometry}
\end{figure}
\begin{equation}\label{ns0.2}
\begin{cases}
\partial_{t} u - \Delta u + \nabla p_0 = 0& \text{in} \ (0,T)\times \Omega,\\
\div u=0& \text{in} \ (0,T)\times \Omega,\\
u=0 & \text{on} \ (0,T)\times\Gamma_0,\\
u=\partial_{t}\eta e_2 & \text{on} \ (0,T)\times \Gamma_1,\\
\partial_{t}^{2} \eta +\partial_{x_{1}}^{4}\eta -\partial^{2}_{x_1} \eta -\partial_{t}\partial_{x_1}^{2} \eta = -\mathbb{T}(u,p_0)n_{|\Gamma_1}\cdot e_2 & \text{in}\ (0,T)\times \mathcal{I},\\
u(0,\cdot\,)=u^0 & \text{in} \ \Omega, \\
\eta(0,\cdot\,)=\eta_1^0, \quad \partial_t \eta(0,\cdot\,)=\eta_1^0 & \text{in} \ \mathcal{I}.
\end{cases}
\end{equation}
Before writing the corresponding observability inequality, let us mention an important remark and introduce some notation. We set
\[
L^2_0(\mathcal{I}):=\left\{ f\in L^2(\mathcal{I}) \ ; \ \int_0^{2\pi} f(x_1)\ \dx_1=0\right\}.
\]
\begin{rema}
Using the particular geometry considered here, we can simplify the above adjoint system. First on $\Gamma_1$, $n=e_2$ and using~\eqref{ws4.6}, we deduce
\begin{equation}\label{ns0.3-T}
-\mathbb{T}(u,p_0)n\cdot e_2= -2\partial_{x_2} u_2 +p_0=2\partial_{x_1} u_1 +p_0=p_0 \quad \text{on} \ \Gamma_1,
\end{equation}
since $u_1(x_1,1)=0$ for $x_1\in \mathcal{I}$.

Moreover, using the incompressibility of the fluid and the boundary conditions, we deduce~that
\[
0=\int_\Omega \div u \ \dx =\frac{\dd}{\dt}\int_{0}^{2\pi} \eta \ \dx_1.
\]
Assuming that $\eta_1^0\in L^2_0(\mathcal{I})$ then, we deduce that for all $t\geq 0$, $\eta(t,\cdot\,)\in L^2_0(\mathcal{I})$. Using this condition on the beam equation leads to the following condition on the pressure:
\begin{equation}\label{meanpressure}
\int_{0}^{2\pi} p_0(t,x_1,1)\ \dx_1=0.
\end{equation}
In particular, in contrast with the standard Stokes system, the pressure is not determined up to a constant.
\end{rema}

We define the operators associated with the beam equation:
\begin{align}\label{defA1}
&&\mathcal{D}(A_1)&:=H^4(\mathcal{I})\cap L^2_0(\mathcal{I}), & A_1 \eta&:=\partial_{x_1}^4 \eta-\partial_{x_1}^2 \eta,&&
\\\label{defA2}
&&\mathcal{D}(A_2)&:=H^2(\mathcal{I})\cap L^2_0(\mathcal{I}), & A_2 \eta&:=-\partial_{x_1}^2 \eta.
\end{align}
We also define the Hilbert space of states for our system:
\begin{equation}\label{defcalH}
\mathcal{H}:=\left\{ (u,\eta_1,\eta_2)\in L^2(\Omega)\times \mathcal{D}(A_1^{1/2})\times L^2_0(\mathcal{I}) \ ; \ u_2= \eta_2 \ \text{on} \ \Gamma_1, \ u_2=0 \ \text{on} \ \Gamma_0,\ \div u=0 \ \text{in} \ \Omega
\right\},
\end{equation}
endowed with the canonical scalar product of $L^2(\Omega)\times \mathcal{D}(A_1^{1/2})\times L^2(\mathcal{I})$. With the above remark and notation, the adjoint system writes
\begin{equation}\label{ns0.3}
\begin{cases}
\partial_{t} u - \Delta u + \nabla p_0 = 0& \text{in} \ (0,T)\times \Omega,\\
\div u=0& \text{in} \ (0,T)\times \Omega,\\
u=0 & \text{on} \ (0,T)\times\Gamma_0,\\
u=\partial_{t}\eta e_2 & \text{on} \ (0,T)\times \Gamma_1,\\
\partial_{t}^{2} \eta +A_1\eta +A_2\partial_{t}\eta = {p_0}_{|_{\Gamma_{1}}} & \text{in}\ (0,T)\times \mathcal{I},\\
u(0,\cdot\,)=u^0 &\text{in} \ \Omega, \\
\eta(0,\cdot\,)=\eta_1^0, \quad \partial_t \eta(0,\cdot\,)=\eta_2^0 & \text{in} \ \mathcal{I},
\end{cases}
\end{equation}
with the condition~\eqref{meanpressure}. Our main result stated below is an observability inequality for~\eqref{ns0.3}:

\begin{theo}\label{T01}
Assume $T>0$, $\omega\Subset \Omega$ and $\mathcal{J}\Subset \mathcal{I}$ are nonempty open sets. For any $[u^0, \eta_1^0, \eta_2^0]\in \mathcal{H}$, the solution of~\eqref{ns0.3} satisfies
\begin{multline}\label{di17:19}
\left\| u(T,\cdot\,)\right\|_{L^2(\Omega)}^2 + \left\| \eta(T,\cdot\,)\right\|_{H^2(\mathcal{I})}^2+ \left\| \partial_t \eta(T,\cdot\,)\right\|_{L^2(\mathcal{I})}^2
\\
\leq k_T^2\left(
\iint_{(0,T)\times \omega} \left| u \right|^2 \ \dx \ \dt +
\iint_{(0,T)\times \mathcal{J}} \left| \partial_t \eta \right|^2 \ \dx_1 \ \dt
\right),
\end{multline}
and we can choose $k_T$ in the form
\begin{equation}\label{je11:14}
k_T = C e^{C/T^2},
\end{equation}
with a constant $C>0$.
\end{theo}

The controllability of fluid-structure interaction systems has already been tackled in the case where the structure is a rigid body in~\cite{MR3085093, MR2375750, MR2139944, MR2317341, MR4043319}. Up to our knowledge, the above theorem is the first result of controllability for the system~\eqref{ns0.1}. Let us mention also~\cite{Sourav2} where the author obtains an observability inequality for the adjoint of a linearized simplified compressible fluid-structure model similar to our system.

Let us point out that due to the structural damping in the beam equation ($-\partial_{t}\partial_{x_1}^2 \zeta$) the corresponding beam equation becomes a parabolic equation (see, for instance, \cite{CheTri1989a}). In a previous work~\cite{buffe:hal-03331176}, we have replaced the damped beam equation by a heat equation and we have shown the corresponding controllability result. The proof done here is inspired by our previous work, and in particular, in the proof of the observability, we first apply results on the heat equations to the fluid velocity by considering the pressure as a source term, (in the spirit of~\cite{fernandez2004local}). Then, we estimate the pressure by using that it satisfies a Laplace equation. Since the boundary conditions of this Laplace equation are difficult to handle, our estimates on the pressure depend on the boundary value of the pressure and more precisely on the high frequencies of the pressure on the boundary of the fluid domain. To conclude, we apply some energy inequalities combined with a high frequency argument in the horizontal direction to estimate these high frequencies. Using the microlocal analysis near boundaries and interfaces to derive Carleman estimates and to show the controllability of coupled parabolic systems is quite standard and one can quote for instance~\cite{BelLeRou1, BelLeRou2, Buffe17, LRLR, LeRouLerner, LRR, LRRparabolic} and the recent books~\cite{CarlemanBook1,Carlemanbook2} for elliptic counterparts).

One of the main differences with~\cite{buffe:hal-03331176} is that we work here directly with the time variable whereas in the previous work we show a spectral inequality and then use an abstract method (\cite{Leautaud, LebeauRobbiano}) to deduce the corresponding observability inequality. Here we do not follow the same approach since it uses that the main operator of our system is self-adjoint, and here our main operator is not self-adjoint or even a perturbation of a self-adjoint operator as in the framework considered in~\cite{Leautaud}. A consequence of working directly with the time variable is that the separation between low and high frequencies is done through a pseudo-differential operator, which symbol depends on time, and in particular we need some standard commutator estimates from these operators in order to handle the high frequencies.

\begin{rema}
With respect to~\cite{buffe:hal-03331176} or to the stabilization result~\cite{MR3619065}, one should expect to obtain the controllability of~\eqref{tak2.3} or of~\eqref{ns0.1} without any control on the beam equation ($g\equiv 0$). However, with our present approach, it seems difficult to handle the elastic displacement without any observation on the beam equation. Even with the presence of two controls, a particular treatment of the coupling between the pressure and the elastic displacement in the proof of the observability is needed. Concerning the particular geometry, we are using it several times in order to simplify several proofs but the corresponding result in a general geometry should hold even if it is not a direct consequence of our work.
\end{rema}

We deduce from \cref{T01} the local controllability of~\eqref{tak2.3}:

\begin{theo}\label{T02}
Assume $T>0$ and that $\omega\Subset \Omega$ and $\mathcal{J}\Subset \mathcal{I}$ are nonempty open sets. There exists $R_0>0$ such that for any $\zeta_1^0\in \mathcal{D}(A_1^{3/4})$, $\zeta_2^0\in \mathcal{D}(A_1^{1/4})$, $w^0\in H^1(\Omega_{\zeta_1^0})$ satisfying
\begin{equation}\label{jeu14:22}
\div w^0=0\ \text{in}\ \Omega,\quad w^0=0 \ \text{on} \ \Gamma_0,\quad w^0(x_1,1+\zeta_1^0(x_1))=\zeta_2^0(x_1) e_2 \quad (x_1\in \mathcal{I}),
\end{equation}
and
\begin{equation}\label{jeu14:27}
\left\| \zeta_1^0\right\|_{H^3(\mathcal{I})}+\left\|\zeta_2^0\right\|_{H^1(\mathcal{I})}+ \left\| w^0 \right\|_{H^1\left(\Omega_{\zeta_1^0}\right)} \leq R_0,
\end{equation}
there exists a control
\[
(f,g)\in L^2(0,T;L^2(\omega))\times L^2(0,T;L^2(\mathcal{J}))
\]
such that the solution of~\eqref{tak2.3} satisfies
\[
\zeta(T,\cdot\,)=0, \quad \partial_t \zeta(T,\cdot\,)=0 \quad \text{in} \ \mathcal{I},\quad w(T,\cdot\,)=0 \quad \text{in} \ \Omega.
\]
\end{theo}

The proof of \cref{T02} is quite standard from \cref{T01}: we need to estimate the coefficients of the change of variables and use a fixed point argument. Similar procedure is done to show the well-posedness or the stabilization of the system. We only sketch the proof of \cref{T02}, the details can be found for instance in~\cite{MR2745779, MR3619065}.

The outline of the article is as follows: in the next section, we complete the functional setting needed in this article, introduce the Carleman weights and some classical results on pseudodifferential operators. \cref{sec_car} is devoted to Carleman estimates: a Carleman estimate for the heat equation, a Carleman estimate for the damped beam equation and a Carleman estimate for the pressure. Gathering them yields an estimate of the fluid velocity and pressure and of the elastic displacement by terms localized in $\omega$ or in $\mathcal{J}$ and by high frequencies of the pressure on the boundary. To get rid of these last terms, we show in \cref{sec_hf} high frequency estimates using the Stokes system. This allows us to show the observability inequality in \cref{sec_obs}. We give the sketch of the proof of \cref{T02} in \cref{sec_proof}. Finally, in \cref{sec_tec}, we recall some technical results concerning the Carleman estimates of \cref{sec_car}.

\begin{nota}
In the whole paper, we use $C$ as a generic positive constant that does not depend on the other terms of the inequality. The value of the constant $C$ may change from one appearance to another. We also use the notation $X\lesssim Y$ if there exists a constant $C>0$ such that we have the inequality $X\leq CY$.
\end{nota}



\section{Notation and preliminaries}
\subsection{Functional setting}\label{sec_func}
We complete the notation introduced in the introduction: we consider the control operator for the beam equation:
\[
B_\mathcal{J}g:=P_{L^2_0(\mathcal{I})} \left(1_{\mathcal{J}} g\right),
\]
where $P_{L^2_0(\mathcal{I})} : L^2(\mathcal{I}) \to L^2_0(\mathcal{I})$ is the orthogonal projection. With the above notation and~\eqref{defA1}, \eqref{defA2}, the beam equation in~\eqref{ns0.1} writes
\[
\partial_{t}^2 \zeta+A_1 \zeta + A_2 \partial_t \zeta=P_{L^2_0(\mathcal{I})}\pi +B_\mathcal{J}g.
\]
We also consider the orthogonal projection on the space $\mathcal{H}$ defined by~\eqref{defcalH}:
\[
\mathcal{P} : L^2(\Omega)\times \mathcal{D}(A_1^{1/2})\times L^2_0(\mathcal{I}) \to \mathcal{H}.
\]
We recall (see, for instance, \cite[Proposition~3.1]{MR3619065}) that the orthogonal of $\mathcal{H}$ in $L^2(\Omega)\times \mathcal{D}(A_1^{1/2})\times L^2_0(\mathcal{I})$ is given by
\begin{equation}\label{09:07}
\mathcal{H}^{\perp}=\left\{ (\nabla p, 0, P_{L^2_0(\mathcal{I})} p_{|\Gamma_1}) \ ; \ p\in H^1(\Omega) \right\}.
\end{equation}
Then we define the space
\[
\mathcal{V}:=\left\{ (u,\eta_1,\eta_2)\in H^1(\Omega)\times \mathcal{D}(A_1^{3/4})\times \mathcal{D}(A_1^{1/4}) \ ; \ u= \eta_2e_2 \ \text{on} \ \Gamma_1, \quad u=0 \ \text{on} \ \Gamma_0,
\quad \div u=0 \ \text{in} \ \Omega
\right\},
\]
and the unbounded operator $\mathcal{A}$ associated with~\eqref{ns0.1}:
\[
\mathcal{D}(\mathcal{A}):=\mathcal{V}\cap \left[H^2(\Omega)\times \mathcal{D}(A_1)\times \mathcal{D}(A_1^{1/2})\right],
\quad
\mathcal{A}
\begin{bmatrix}
u\\ \eta_1 \\ \eta_2
\end{bmatrix} :=\mathcal{P}
\begin{bmatrix}
\Delta u\\ \eta_2 \\ -A_1 \eta_1 -A_2 \eta_2
\end{bmatrix}.
\]
It is shown (see, for instance, \cite[Proposition~3.11]{MR3619065}) that $\mathcal{A}$ is the infinitesimal generator of an analytic semigroup on $\mathcal{H}$. We have in particular that if $F\in L^2(0,T;\mathcal{H})$, $\Phi^0\in \mathcal{V}$, then there exists a unique solution
\[
\Phi\in L^2(0,T;\mathcal{D}(\mathcal{A})) \cap C^0([0,T];\mathcal{V}) \cap H^1(0,T;\mathcal{H})
\]
to
\begin{equation}\label{ma11:09}
\frac{\dd\Phi}{\dt}=\mathcal{A}\Phi +F \quad \text{in} \ (0,T), \quad \Phi(0)=\Phi^0
\end{equation}
and we have the estimate
\begin{multline}\label{ma11:10}
\left\| \Phi\right\|_{L^2(0,T;H^2(\Omega)\times \mathcal{D}(A_1)\times \mathcal{D}(A_1^{1/2}))} +\left\| \Phi\right\|_{H^1(0,T;L^2(\Omega)\times \mathcal{D}(A_1^{1/2})\times L^2(\mathcal{I}))}
\\
\lesssim
\left\| F\right\|_{L^2(0,T;L^2(\Omega)\times \mathcal{D}(A_1^{1/2})\times L^2(\mathcal{I}))} +\left\| \Phi^0 \right\|_{\mathcal{V}}.
\end{multline}

Finally, we consider the control operator:
\[
\mathcal{B}
\begin{bmatrix}
f\\ g
\end{bmatrix} :=\mathcal{P}
\begin{bmatrix}
1_\omega f\\ 0 \\ B_\mathcal{J}g
\end{bmatrix}.
\]
Using the above notation and~\eqref{09:07}, we can write~\eqref{ns0.1} as
\begin{equation}\label{di16:08}
\frac{\dd}{\dt}
\begin{bmatrix}
w\\ \zeta \\ \partial_t \zeta
\end{bmatrix} =\mathcal{A}
\begin{bmatrix}
w\\ \zeta \\ \partial_t \zeta
\end{bmatrix} +\mathcal{B}
\begin{bmatrix}
f\\ g
\end{bmatrix}
\quad \text{in} \ (0,T), \quad
\begin{bmatrix}
w\\ \zeta \\ \partial_t \zeta
\end{bmatrix}(0)=
\begin{bmatrix}
w^0\\ \zeta_1^0 \\ \zeta_2^0
\end{bmatrix}.
\end{equation}
We say that the above system is null-controllable in time $T>0$ if for any $
\begin{bmatrix}
w^0,\ \zeta_1^0, \ \zeta_2^0
\end{bmatrix}\in \mathcal{H}$, there exists a control $
\begin{bmatrix}
f,\ g
\end{bmatrix}\in L^2(0,T;L^2(\omega)\times L^2(\mathcal{J})) $ such that the solution of the above system satisfies
\[
\begin{bmatrix}
w\\ \zeta \\ \partial_t \zeta
\end{bmatrix}(T)=0.
\]
A classical result (see, for instance, \cite[Theorem~11.2.1, p.~357]{TucsnakWeiss}) states that the null-controllability is equivalent to the final-state observability of the adjoint system: there exists $k_T>0$ such that for any $
\begin{bsmallmatrix}
u^0\\ \eta_1^0 \\ \eta_2^0
\end{bsmallmatrix}\in \mathcal{H}$, the solution of
\begin{equation}\label{di16:32}
\frac{\dd}{\dt}
\begin{bmatrix}
u\\ \eta_1 \\ \eta_2
\end{bmatrix} =\mathcal{A}^*
\begin{bmatrix}
u\\ \eta_1 \\ \eta_2
\end{bmatrix}\quad \text{in} \ (0,T), \quad
\begin{bmatrix}
u\\ \eta_1 \\ \eta_2
\end{bmatrix}(0)=
\begin{bmatrix}
u^0\\ -\eta_1^0 \\ \eta_2^0
\end{bmatrix}
\end{equation}
satisfies
\begin{equation}\label{di16:36}
\left\|
\begin{bmatrix}
u\\ \eta_1 \\ \eta_2
\end{bmatrix}(T) \right\|_{\mathcal{H}}^2 \leq k_T^2 \int_0^T \left\| \mathcal{B}^*
\begin{bmatrix}
u\\ \eta_1 \\ \eta_2
\end{bmatrix}(t)\right\|_{L^2(\omega)\times L^2(\mathcal{J})}^2 \ \dt.
\end{equation}
One can show that
\[
\mathcal{D}(\mathcal{A}^*)=\mathcal{D}(\mathcal{A}),
\quad
\mathcal{A}^*
\begin{bmatrix}
u\\ \eta_1 \\ \eta_2
\end{bmatrix} =\mathcal{P}
\begin{bmatrix}
\Delta u\\ -\eta_2 \\ A_1 \eta_1 -A_2 \eta_2
\end{bmatrix}
\]
and
\[
\mathcal{B}^*
\begin{bmatrix}
u\\ \eta_1 \\ \eta_2
\end{bmatrix} =
\begin{bmatrix}
u_{|\omega}\\ {\eta_2}_{|\mathcal{J}}
\end{bmatrix}.
\]
Setting $\eta=-\eta_1$ we see that~\eqref{di16:32} writes as~\eqref{ns0.3} or in the following abstract form
\begin{equation}\label{0907}
\frac{\dd}{\dt}
\begin{bmatrix}
u\\ \eta \\ \partial_t \eta
\end{bmatrix} =\mathcal{A}
\begin{bmatrix}
u\\ \eta \\ \partial_t \eta
\end{bmatrix}\quad \text{in} \ (0,T), \quad
\begin{bmatrix}
u\\ \eta \\ \partial_t \eta
\end{bmatrix}(0)=
\begin{bmatrix}
u^0\\ \eta_1^0 \\ \eta_2^0
\end{bmatrix}.
\end{equation}
The observability inequality~\eqref{di16:36} writes as~\eqref{di17:19}.

\subsection{Weight functions for the Carleman estimates}\label{Weight functions}

We consider nonempty open subsets $\mathcal{J}_0\Subset \mathcal{J}$ and $\omega_0\Subset \omega$ and (see, for instance, \cite[Lemma~1.1]{FursikovImanuvilov}, \cite[Theorem~9.4.3]{TucsnakWeiss}) two smooth functions $\psi_{\mathcal{I}}$ and $\psi_{\Omega}$ satisfying
\begin{gather}\label{rt1054}
\psi_{\mathcal{I}}>0 \ \text{in} \ \mathcal{I}, \quad
\psi_{\mathcal{I}}'(x_{1})=0 \Rightarrow x_{1}\in \mathcal J_0,
\\
\label{rt1055}
\psi_{\Omega}>0 \ \text{in} \ \Omega, \quad \psi_{\Omega}=0 \ \text{and} \ \partial_n \psi_{\Omega}=-1 \ \text{on} \ \partial \Omega, \quad
\nabla \psi_{\Omega}(x) =0 \Rightarrow x\in \omega_0,
\end{gather}
with
\begin{equation}\label{ve17:43}
\mathcal J_0 \Subset \mathcal J, \quad \omega_0 \Subset \omega.
\end{equation}
In fact, using our particular geometry, one can show directly the existence of such functions $\psi_{\mathcal{I}}$ and $\psi_{\Omega}$. We set $\psi_1(x_1):=2+\sin(x_1)$ and we consider $\psi_2\in C^\infty([0,1])$,
%odd, $\psi_2(x_2)=x_2$ in a neighborhood of $0$,
$\psi_2(x_2)=x_2$ in a neighborhood of $0$, $\psi_2(x_2)=1-x_2$ in a neighborhood of $1$
and $\psi_2'(x_2)=0 \Leftrightarrow x_2=1/2$. We also consider $\theta\in C^\infty(\mathbb{R})$ with compact support in $(0,1)$ and such that $\theta \equiv 1$ in a neighborhood of $1/2$. Then for $\varepsilon>0$ small enough,
\[
\widetilde{\psi_\Omega}(x_1,x_2)=\psi_2(x_2)+\varepsilon \theta(x_2) \psi_1(x_1)
\]
satisfies $\widetilde{\psi_{\Omega}}>0$ in $\Omega,$ $\widetilde{\psi_{\Omega}}=0$ and $\partial_n \widetilde{\psi_{\Omega}}=-1$ on $\partial \Omega$ and it has only two critical points: $(\pi/2,1/2)$ and $(-\pi/2,1/2)$. By a change of variables on $\psi_1$ and on $\widetilde{\psi_{\Omega}}$ (see, for instance, \cite[Proposition~14.3.1]{TucsnakWeiss}), we obtain functions $\psi_{\mathcal{I}}$ and $\psi_{\Omega}$ satisfying~\eqref{rt1054} and~\eqref{rt1055}.

We also denote by $\ell$ the function defined by
\begin{equation}\label{1131119}
\ell(t):=t(T-t).
\end{equation}
Let us consider $\Psi:=\left\|\psi_{\mathcal{I}}\right\|_{L^\infty(\mathcal{I})}+\left\|\psi_{\Omega}\right\|_{L^\infty(\Omega)}$ and for $\lambda\geq \mu>0$, let us define the following functions
\begin{align}\label{1131123}
\varphi(t,x_1,x_2) &:= \frac{1}{\ell(t)^2}(e^{\lambda\psi_\Omega(x_1,x_2)+\mu \psi_{\mathcal{I}}(x_1)+8\lambda\Psi}-e^{10\lambda\Psi}),&
\xi(t,x_1,x_2)&:=\frac{1}{\ell(t)^2}e^{\lambda\psi_\Omega(x_1,x_2)+\mu \psi_{\mathcal{I}}(x_1)+8\lambda\Psi},
\\
\label{1131123bis}
\varphi_0(t,x_1) &:= \frac{1}{\ell(t)^2}(e^{\mu \psi_{\mathcal{I}}(x_1)+8\lambda\Psi}-e^{10\lambda\Psi}),&
\xi_0(t,x_1)&:=\frac{1}{\ell(t)^2}e^{\mu \psi_{\mathcal{I}}(x_1)+8\lambda\Psi}.
\end{align}
We also define for $\lambda\geq \mu>0$ the function
\begin{equation}\label{defpsi}
\psi(x_{1},x_{2}):=\frac{\mu}{\lambda}\psi_{\mathcal{I}}(x_{1})+\psi_{\Omega}(x_{1},x_{2}).
\end{equation}

\subsection{Spatial truncation}\label{sec_trunc}
In order to use pseudodifferential operators in the $x_1$ variables, we consider that our functions are $2\pi$-periodic functions defined in the domains
\[
\Omega^\infty:=\mathbb{R}\times (0,1), \quad \Gamma_0^\infty:=\mathbb{R}\times \{0\}, \quad \Gamma_1^\infty:=\mathbb{R}\times \{1\}.
\]
In the adjoint system~\eqref{ns0.3}, we also replace the pressure $p_0$ that satisfies~\eqref{meanpressure} by a pressure $p$ satisfying another condition. More precisely, we consider $\omega_1$ an open set such that $\omega_0\Subset \omega_1 \Subset \omega$ and we define
\[
c_p(t):=-\int_{\omega_1} p_0(t,x) \ \dx
\]
and
\begin{equation}\label{pcp}
p:=p_0+c_p.
\end{equation}
Then the pressure $p$ verifies the condition
\begin{equation}\label{ma10:17}
\int_{\omega_1} p(t,x) \ \dx =0 \quad \text{in} \ (0,T).
\end{equation}
We consider $\chi^\infty\in C^\infty(\mathbb{R};[0,1])$ with compact support and such that $\chi^\infty \equiv 1$ in $[0,2\pi]$. We set
\begin{equation}\label{utrunc}
u^\infty:=\chi^\infty u, \quad p^\infty:=\chi^\infty p, \quad \eta^\infty:=\chi^\infty \eta.
\end{equation}
Then we deduce from~\eqref{ns0.3} that
\begin{equation}\label{ns0.31}
\begin{cases}
\partial_{t} u^\infty - \Delta u^\infty + \nabla p^\infty = f^{(1)}& \text{in} \ (0,T)\times \Omega^\infty,\\
\div u^\infty= f^{(2)}& \text{in} \ (0,T)\times \Omega^\infty,\\
u^\infty=0 & \text{on} \ (0,T)\times\Gamma_0^\infty,\\
u^\infty=\partial_{t}\eta^\infty e_2 & \text{on} \ (0,T)\times \Gamma_1^\infty,
\end{cases}
\end{equation}
and
\begin{equation}\label{ns0.32}
\Delta p^\infty= f^{(3)}\quad \text{in} \ (0,T)\times \Omega^\infty,
\end{equation}
where
\begin{equation}\label{1519}
f^{(1)}:=-\left(\chi^\infty\right)'' u - 2\left(\chi^\infty\right)' \partial_{x_1} u + \left(\chi^\infty\right)' p e_1,
\quad f^{(2)}:=\left(\chi^\infty\right)' u_1,
\quad f^{(3)} = (\chi^{\infty})'' p+ 2(\chi^{\infty})'\partial_{x_{1}} p.
\end{equation}


\subsection{Pseudodifferential operators} We consider a parameter $\tau\geq 1$ and an order function
\begin{equation}\label{defLambda}
\Lambda_{\tau}(k):=\sqrt{\tau^2+k^2} \quad (k\in \mathbb{R}),
\end{equation}
where $k$ corresponds to the Fourier variable associated with $x_1$. For $m\in \mathbb{R}$, we denote by $\Sbf^{m}_{\tau}$ the space of complex smooth functions $a=a(x_1,k,\tau)$ defined on $\mathbb{R}\times \mathbb{R}\times [1,\infty)$ and such that for all $\alpha, \beta\in \mathbb{N}$ there exists $C_{\alpha,\beta}>0$
\begin{equation}\label{Smtau}
\left|\partial^{\alpha}_{x_1}\partial_{k}^{\beta}a(x_{1},k,\tau)\right|\leq C_{\alpha,\beta} \Lambda_{\tau}^{m-\beta}(k) \quad ((x_1,k,\tau)\in \mathbb{R}\times \mathbb{R}\times [1,\infty)).
\end{equation}
For instance, we have $\Lambda_\tau^m \in \Sbf^{m}_{\tau}$ and for any $C\in \mathbb{R}$, the function
\[
(k,\tau) \mapsto \frac{\tau^2-C k^2}{\tau^2+k^2}
\]
is in $\Sbf^{0}_{\tau}$. We also recall the following classical lemma (see, for instance, \cite[Proposition~2.3]{CarlemanBook1} or~\cite[p.~73, Lemma~18.1.10]{Hormander3} in the classical setting)

\begin{lemm}\label{L01}
If $a\in \Sbf^{0}_{\tau}$ and $\chi_0\in C^{\infty}(\mathbb{R})$. Then $\chi_0(a)\in\Sbf^{0}_{\tau}$.
\end{lemm}

From $a\in \Sbf^{m}_{\tau}$, we can define the following operator on the Schwartz space on $\mathbb{R}:$
\[
\left[\Op(a) u \right](x_1):=\frac{1}{2\pi} \iint_{\mathbb{R}^2} e^{ik(x_1-y_1)} a(x_1,k,\tau) u(y_1) \ \dy_1 \dk.
\]
We can also extend this operator to the Schwartz space on $[0,T] \times \mathbb{R}\times [0,1]$ by a similar formula:
\[
\left[\Op(a) u \right](t,x_1,x_2):=\frac{1}{2\pi} \iint_{\mathbb{R}^2} e^{ik(x_1-y_1)} a(x_1,k,\tau) u(t,y_1,x_2) \ \dy_1 \dk.
\]
From symbolic calculus, we have the following results (see, for instance, \cite[pp.~27-28, Theorem~2.22 and Corollary~2.23]{CarlemanBook1})

\begin{theo}\label{Commutatorcorollary}
Let $m,m'\in\mathbb R$ and let $a\in \Sbf^{m}_{\tau}$, $b\in\Sbf^{m'}_{\tau}$. Then there exist $c\in \Sbf^{m+m'}_{\tau}$ and $d\in \Sbf^{m+m'-1}_{\tau}$ such that
\[
\Op(a)\circ\Op(b) = \Op(c), \quad [\Op(a),\Op(b)] = \Op(d).
\]
\end{theo}

We can extend the operator associated with a symbol of order $m$ to Sobolev spaces. For instance we have the following result (see~\cite[p.~29, Theorem~2.26]{CarlemanBook1})

\begin{theo}\label{Continuitytheorem}
Let $m,m'\in\mathbb R$, and let $a\in\Sbf^{m}_{\tau}$. Then, $\Op(a) : H^{m+m'}(\mathbb{R}) \to H^{m'}(\mathbb{R})$ and if $m,m'\in\mathbb N$, we have
\[
\sum_{i+j\leq m'} \tau^{2i} \left\| \partial^{j}_{x_{1}}\Op(a) u \right\|_{L^{2}(\mathbb R)}^{2} \lesssim
\sum_{i+j\leq m+m'} \tau^{2i} \left\| \partial^{j}_{x_{1}} u \right\|_{L^{2}(\mathbb R)}^{2}.
\]
\end{theo}

In what follows, we assume that the parameter $\tau$ is related to functions defined in \cref{Weight functions} through the formula
\begin{equation}\label{deftau}
\tau:=\tau(t)= \frac{s\lambda e^{8\lambda \Psi}}{\ell^2(t)}.
\end{equation}
In particular, $\tau$ is a function of time and there exist $s_0>0$ and $\lambda_0>0$ such that if $s\geq s_0 T^4$ and $\lambda\geq \lambda_0$, then
\begin{equation}\label{lu11:34}
\tau\geq \frac{\tau}{\lambda} \geq 1.
\end{equation}
\begin{rema}
Due to~\eqref{deftau}, the symbols in $\Sbf^{m}_{\tau}$ depends on the time variable through the parameter $\tau$. The continuity estimates of \cref{Continuitytheorem} are uniform with respect to $\tau$, and thus with respect to the time variable if it only appears in the parameter $\tau$. In what follows, some symbols may depend on time, but not as a function of $\tau$, this occurs for instance when considering $\partial_{t}\tau$. In that case, we always decompose such symbols in terms of the form $b(t)a(x_{1},k,\tau)$ where $b$ is a bounded function of time, and $a\in\Sbf^{m}_{\tau}$.
\end{rema}

An important example of symbol used in what follows is a function of the form
\[
\chi(\tau,k):=\chi_0\left(\frac{\tau^2-C k^2}{\tau^2+k^2} \right),
\]
where $C$ is a constant and $\chi_0\in C^\infty(\mathbb{R})$. From \cref{L01}, we have that $\chi\in \Sbf^{0}_{\tau}$ and one can check that
\[
[\partial_{x_{1}}, \Op(\chi)] = [\partial_{x_{2}}, \Op(\chi)]=0.
\]
Moreover, we have the following result on the time derivative of $\chi$:

\begin{lemm}\label{L02}
Let $\chi$ be defined as above. Then
\[
\partial_{t}\chi \in \frac{\ell'}{(\lambda s)^{1/2} }\tau^{\frac{5}{2}}\Sbf^{-2}_{\tau}.
\]
\end{lemm}
\begin{proof}
By standard computations and~\eqref{deftau},
\begin{equation}\label{di19:48}
\partial_t \chi(\tau, k)=\chi_0'\left(\frac{\tau^{2} -C k^{2}}{\tau^{2}+k^{2}}\right)\frac{2\left(C+1 \right) k^2 \tau \partial_t \tau}{\left(\tau^{2}+k^{2}\right)^2},
\quad
\partial_t \tau= -2\ell' \frac{s\lambda e^{8\lambda \Psi}}{\ell^{3}}.
\end{equation}
We have
\[
\frac{s\lambda e^{8\lambda \Psi}}{\ell^{3}} \leq \frac{\tau^{3/2}}{\left(s\lambda\right)^{1/2}}
\]
so that using \cref{L01} and \cref{Commutatorcorollary}, we deduce the result.
\end{proof}



\section{Carleman estimates}\label{sec_car}
In this section, we show a Carleman estimate for the solutions of~\eqref{ns0.2}. Using the weights introduced in \cref{Weight functions}, we define the following weighted integrals:
\begin{multline}\label{lu10:26}
I_1(s,\lambda,\eta):=
\lambda\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} \left(s^{10} \xi_{0}^{10} |\eta |^{2} +s^{8} \xi_{0}^{8} |\partial_{x_1} \eta |^{2} +s^{6} \xi_{0}^{6} \left(| \partial_{x_1}^2\eta |^{2} +| \partial_{t} \eta |^{2} \right)
\right)
\ \dt\, \dx_1
\\
\begin{aligned}[b]
&+\lambda\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} s^{4}\xi_{0}^{4} \left(| \partial_{x_1}^{3}\eta |^{2}+ |\partial_{x_1} \partial_{t}\eta |^{2}\right)
\ \dt\, \dx_1
\\
&+\lambda\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} s^2\xi_{0}^2 \left(| \partial_{x_1}^{4}\eta |^{2}+ |\partial_{t}\partial_{x_1}^2\eta |^{2}+ |\partial_{t}^2\eta |^{2}\right)
\ \dt\, \dx_1
\\
&+\lambda\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} \left(
\left| \partial_{x_1}^{5}\eta \right|^{2}+ | \partial_{t}^2\partial_{x_1}\eta |^{2}+ | \partial_{t}\partial_{x_1}^3 \eta |^{2}
\right)
\ \dt\, \dx_1,
\end{aligned}
\end{multline}
\begin{multline}\label{lu10:27}
I_2(s,\lambda,u):=
\iint_{(0,T)\times \Omega} \lambda^{2} \left(|\nabla^{2}u|^{2}+(\partial_t u)^2 \right) e^{2s\varphi} \ \dt\,\dx +\iint_{(0,T)\times \Omega} s^2\lambda^{4} \xi^{2} e^{2s\varphi} | \nabla u |^{2} \ \dt\,\dx
\\
+ \iint_{(0,T)\times \Omega} s^{4}\lambda^{6}\xi^{4} e^{2s\varphi} |u|^{2} \ \dt\,\dx,
\end{multline}
and
\begin{multline}\label{lu10:28}
I_3(s,\lambda,p^\infty):=
\iint_{(0,T)\times \Omega^{\infty}} s^{3}\lambda^{4}\xi^{3} e^{2s\varphi} |p^{\infty}|^{2} \ \dt\,\dx +\iint_{(0,T)\times \Omega^{\infty}} s \lambda^2 \xi e^{2s\varphi} |\nabla p^{\infty} |^{2}\ \dt\,\dx
\\
+\iint_{(0,T)\times \partial\Omega^\infty} s^3 \lambda^3 \xi_0^3 e^{2s\varphi_0} \left| p^{\infty} \right|^{2} \ \dt\, \dx_1 +\iint_{(0,T)\times \partial\Omega^\infty} s \lambda \xi_0 e^{2s\varphi_0} \left| \partial_{x_1} p^{\infty} \right|^{2} \ \dt\, \dx_1.
\end{multline}
\begin{rema}
The above quantities depend also on $\mu$ but since we will fix the value of $\mu=\mu_0$ after \cref{carela}, we suppress reference to it in the notation.
\end{rema}

For $\mu_0>1$, we set
\begin{equation}\label{defKK}
K_+:=e^{\mu_0 \max \psi_{\mathcal{I}}}, \quad K_-:=e^{\mu_0 \min \psi_{\mathcal{I}}}.
\end{equation}
In particular, with the definition~\eqref{1131123bis} of $\xi_0$ and the definition~\eqref{deftau} of $\tau$, we have
\begin{equation}\label{sa19:07}
K_- \tau\leq s\lambda \xi_0\leq K_+ \tau.
\end{equation}
Using \cref{L01}, we can define the following symbol of order 0:
\begin{equation}\label{defchi}
\chi(\tau, k):=\chi_0\left(\frac{\tau^{2} -\frac{4 K_{+}}{K_{-}^{3}} k^{2}}{\tau^{2}+k^{2}}\right)\in\Sbf^{0}_{\tau}, \quad
\text{with}\quad
\chi_0 \in C^\infty(\mathbb{R}; [0,1]) \ \text{such that}\
\chi_0=
\begin{cases} 1&\text{in } [3/4,\infty) \\ 0 &\text{in } (-\infty,1/2]
\end{cases}.
\end{equation}

The main result of this section is stated below:

\begin{prop}\label{P04}
Assume $\mathcal{J}_0\Subset \mathcal{J}_1 \Subset \mathcal{J}$ and $\omega_0\Subset \omega_1 \Subset \omega$. There exist $\mu_0>0$, $\lambda_0>0$ and $s_0$ such that for $\mu=\mu_0$, $\lambda\geq \lambda_0$ and $s\geq s_0(T^2+T^4)$, any smooth solution $[u,p_0,\eta]$ of~\eqref{ns0.2} satisfies
\begin{multline}\label{2322}
I_1(s,\lambda,\eta)+I_2(s,\lambda,u)+I_3(s,\lambda,p^\infty)\\
\begin{aligned}[b]
&\lesssim \lambda \iint_{(0,T)\times\mathcal J_1} e^{2s\varphi_{0}}
\left(s^{10}\xi_{0}^{10}|\eta|^{2}+s^{2}\xi_{0}^{2}|\partial_{t}^2\eta|^{2}
\right)\ \dt\, \dx_1
\\
&\qquad+ \iint_{(0,T)\times\omega_1} e^{2s\varphi}\left(s^{4}\lambda^{6}\xi^{4} |u|^{2} \ \dt\,\dx +s^{3}\lambda^{4}\xi^{3} |p^{\infty}|^{2}
\right)\ \dt\, \dx
\\
&\qquad+\iint_{(0,T)\times \partial \Omega^{\infty}} \tau \left| \partial_{x_1}\Op(1-\chi)\left[e^{s\varphi_0} p^{\infty} \right]\right|^2\ \dt\, \dx_1,
\end{aligned}
\end{multline}
where $p^\infty$ is given by~\eqref{pcp} and~\eqref{utrunc}.
\end{prop}

In order to prove \cref{P04}, we first combine a Carleman estimate for the fluid velocity and a Carleman estimate for the elastic deformation (see \cref{carela} and \cref{carvel}). Both estimates contain pressure terms in the right-hand side and to estimates them, we perform a Carleman estimate for the pressure in \cref{carpress}. In this last estimate, we need to put in the right-hand side the trace of the pressure at the boundary, microlocalized in the high frequency regime.

\subsection{A Carleman estimate for the elastic deformation}\label{carela}
In this section, we obtain a Carleman estimate for the elastic deformation, mainly based on the results in~\cite{Sourav}. This is the only part of the work where $\mu\geq \mu_0$, after this, we will fix $\mu=\mu_0$ in the weights $\varphi, \xi, \varphi_0, \xi_0$. To avoid introducing many notations, we keep the same notation $\mu_0$, $s_0$, $\lambda_0$ during the proofs, but their values may change from one appearance to another.

First, we deduce from the definitions~\eqref{1131123bis}, the existence of $\mu_0$ such that for $\lambda\geq \mu\geq \mu_0$, $t\in [0,T]$ and $x_1\in \mathcal{I}$, and $\alpha\geq 0$,
\begin{multline}\label{1700}
\left| \partial_{x_1}^\alpha \varphi_0 \right| +\left| \partial_{x_1}^\alpha \xi_0 \right| \lesssim \mu^\alpha \xi_0 \quad (k\geq 1),
\\
\left| \partial_t \partial_{x_1}^\alpha \varphi_0 \right| +\left| \partial_t \partial_{x_1}^\alpha \xi_0 \right| \lesssim T \mu^\alpha \xi_0^{3/2},
\quad
\left| \partial_t^2 \partial_{x_1}^\alpha \varphi_0 \right| +\left| \partial_t^2 \partial_{x_1}^\alpha \xi_0 \right| \lesssim T^2 \mu^\alpha \xi_0^{2}.
\end{multline}
Moreover, there exists $\mu_0$ such that for $\lambda\geq \mu\geq \mu_0$, for $t\in [0,T]$ and for $x_1\in \mathcal{I}\setminus \mathcal{J}_0$,
\begin{equation}\label{ve17:32}
\mu \xi_0\lesssim \left| \partial_{x_1} \varphi_0 \right|, \quad \mu^2 \xi_0\lesssim \partial_{x_1}^2 \varphi_0.
\end{equation}
With these properties, we can obtain the following result which is proven in~\cite{Sourav}. More precisely, the Carleman estimate below is obtained in~\cite{Sourav} with slightly different weights but the author only uses the above properties in his proof. For sake of completeness, we give in \cref{sec_tecA} a sketch of the corresponding proof.

\begin{theo}\label{CarBeam}
Assume $r\in \mathbb{R}$ and $\mathcal{J}_0\Subset \mathcal{J}_1 \Subset \mathcal{J}$. There exist constants $s_{0}>0$ and $\mu_{0}>0$ such that for any smooth function $\eta$, for any $s\geq s_{0}(T^2+T^4)$, and for any $\lambda \geq \mu\geq \mu_{0}$, we have
\begin{multline}
\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} \left(s^{2r+7}\mu^{2r+8} \xi_{0}^{2r+7} |\eta |^{2} +s^{2r+5}\mu^{2r+6} \xi_{0}^{2r+5} | \partial_{x_1}\eta |^{2}
\right)
\ \dt\, \dx_1
\\
\begin{aligned}[b]
&+\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} s^{2r+3}\mu^{2r+4}\xi_{0}^{2r+3} \left(| \partial_{x_1}^{2}\eta |^{2}+ | \partial_{t}\eta |^{2}\right)
\ \dt\, \dx_1
\\
&+\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} s^{2r+1}\mu^{2r+2}\xi_{0}^{2r+1} \left(| \partial_{x_1}^{3}\eta |^{2}+ |\partial_{t}\partial_{x_1}\eta |^{2}\right)
\ \dt\, \dx_1
\\
&+\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} s^{2r-1}\mu^{2r}\xi_{0}^{2r-1} \left(\left| \partial_{x_1}^{4}\eta \right|^{2}+ | \partial_{t}^2\eta |^{2}+ | \partial_{t}\partial_{x_1}^2 \eta |^{2}\right)
\ \dt\, \dx_1
\end{aligned}
\\
\shoveleft{\lesssim
\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} s^{2r}\mu^{2r}\xi_{0}^{2r} \left| (\partial_{t}^2+\partial_{x_1}^{4}-\partial_{x_1}^{2}-\partial_{t}\partial_{x_1}^2) \eta \right|^{2}\ \dt\, \dx_1}
\\
+ \iint_{(0,T)\times\mathcal J_1} s^{2r+7}\mu^{2r+8}\xi_{0}^{2r+7}e^{2s\varphi_{0}} |\eta|^{2} \ \dt\, \dx_1.
\end{multline}
\end{theo}

As a corollary, we have the following result

\begin{coro}\label{C01}
Assume $\mathcal{J}_0\Subset \mathcal{J}_1 \Subset \mathcal{J}$. There exist constants $s_{0}>0$ and $\mu_{0}>0$ such that for any smooth function $\eta$, for any $s\geq s_{0}(T^2+T^4)$, and for any $\lambda \geq \mu\geq \mu_{0}$, we have
\begin{multline}\label{sourav2}
\begin{aligned}
&\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} \left(s^{10}\mu^{11} \xi_{0}^{10} |\eta |^{2} +s^{8}\mu^{9} \xi_{0}^{8} |\partial_{x_1} \eta |^{2} +s^{6}\mu^{7} \xi_{0}^{6} \left(| \partial_{x_1}^2\eta |^{2} +| \partial_{t} \eta |^{2} \right)\right)\ \dt\, \dx_1
\\
&+\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} s^{4}\mu^{5}\xi_{0}^{4} \left(| \partial_{x_1}^{3}\eta |^{2}+ |\partial_{x_1} \partial_{t}\eta |^{2}\right)\dt\, \dx_1
\\
&+\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} \left(s^2\mu^{3}\xi_{0}^2 \left(| \partial_{x_1}^{4}\eta |^{2}+ |\partial_{t}\partial_{x_1}^2\eta |^{2}+ |\partial_{t}^2\eta |^{2}\right) +\mu \left(\left| \partial_{x_1}^{5}\eta \right|^{2}+ | \partial_{t}^2\partial_{x_1}\eta |^{2}+ | \partial_{t}\partial_{x_1}^3 \eta |^{2}\right)
\right)\dt\, \dx_1
\end{aligned}
\\
\lesssim
\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} s\mu\xi_{0} |\partial_{x_1} (\partial_{t}^2-\partial_{x_1}^{2}+\partial_{x_1}^{4}-\partial_{t}\partial_{x_1}^2) \eta |^{2}\ \dt\, \dx_1
\\
+ \iint_{(0,T)\times\mathcal J_1} e^{2s\varphi_{0}} \left(s^{10}\mu^{11}\xi_{0}^{10}|\eta|^{2}+s^{2}\mu^{3}\xi_{0}^{2}|\partial_{t}^2\eta|^{2}
\right)\ \dt\, \dx_1.
\end{multline}
\end{coro}
\begin{proof}
We first apply \cref{CarBeam} to $\partial_{x_1} \eta$ with $r=1/2$ and with an open set $\mathcal{J}_2$ such that $\mathcal{J}_0\Subset \mathcal{J}_2 \Subset \mathcal{J}_1$ :
\begin{multline}\label{ve19:53}
\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} \left(s^{8}\mu^{9} \xi_{0}^{8} |\partial_{x_1} \eta |^{2} +s^{6}\mu^{7} \xi_{0}^{6} | \partial_{x_1}^2\eta |^{2} + s^{4}\mu^{5}\xi_{0}^{4} \left(| \partial_{x_1}^{3}\eta |^{2}+ | \partial_{t}\partial_{x_1}\eta |^{2}\right)
\right)
\ \dt\, \dx_1
\\
+\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}}
\left(s^{2}\mu^{3}\xi_{0}^{2} \left(| \partial_{x_1}^{4}\eta |^{2}+ |\partial_{t}\partial_{x_1}^2\eta |^{2}\right) +\mu \left(\left| \partial_{x_1}^{5}\eta \right|^{2}+ | \partial_{t}^2\partial_{x_1}\eta |^{2}+ | \partial_{t}\partial_{x_1}^3 \eta |^{2}\right)
\right)
\ \dt\, \dx_1
\\
\lesssim
\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} s\mu\xi_{0} \left| \partial_{x_1}(\partial_{t}^2+\partial_{x_1}^{4}-\partial_{x_1}^{2}-\partial_{t}\partial_{x_1}^2) \eta \right|^{2}\ \dt\, \dx_1
\\
+ \iint_{(0,T)\times\mathcal J_2} s^{8}\mu^{9}\xi_{0}^{8}e^{2s\varphi_{0}} |\partial_{x_1}\eta|^{2} \ \dt\, \dx_1.
\end{multline}
Then, we use a Carleman estimate for the gradient operator (see, for instance, \cite[Lemma~3]{CoronGuerrero}): there exists $s_0>0$ such that for any smooth function $\zeta$, and for any $s\geq s_0 T^4$,
\begin{multline*}
\iint_{(0,T)\times \mathcal{I}} s^{r+2} \mu^{r+3} \xi_0^{r+2} e^{2s\varphi_0} \zeta^2 \ \dt\, \dx_1\\
\lesssim
\iint_{(0,T)\times \mathcal{J}_2} s^{r+2} \mu^{r+3} \xi_0^{r+2} e^{2s\varphi_0} \zeta^2 \ \dt\, \dx_1
+
\iint_{(0,T)\times \mathcal{I}} s^{r} \mu^{r+1} \xi_0^{r} e^{2s\varphi_0} (\partial_{x_1}\zeta)^2 \ \dt\,\dx_1.
\end{multline*}
This Carleman estimate, combined with~\eqref{ve19:53}, yields that for $s\geq s_0(T^2+T^4)$,
\begin{multline}
\begin{aligned}
&\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} \left(s^{10}\mu^{11} \xi_{0}^{10} |\eta |^{2} +s^{8}\mu^{9} \xi_{0}^{8} |\partial_{x_1} \eta |^{2} +s^{6}\mu^{7} \xi_{0}^{6} \left(| \partial_{x_1}^2\eta |^{2} +| \partial_{t} \eta |^{2} \right)
\right)
\ \dt\, \dx_1
\\
&+\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} \left(s^{4}\mu^{5}\xi_{0}^{4} \left(| \partial_{x_1}^{3}\eta |^{2}+ |\partial_{x_1} \partial_{t}\eta |^{2}\right)
\right)
\ \dt\, \dx_1
\\
&+\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} \left(s^2\mu^{3}\xi_{0}^2 \left(| \partial_{x_1}^{4}\eta |^{2}+ |\partial_{t}\partial_{x_1}^2\eta |^{2}+ |\partial_{t}^2\eta |^{2}\right) +\mu \left(\left| \partial_{x_1}^{5}\eta \right|^{2}+ | \partial_{t}^2\partial_{x_1}\eta |^{2}+ | \partial_{t}\partial_{x_1}^3 \eta |^{2}\right)
\right) \dt\, \dx_1
\end{aligned}
\\
\shoveleft{\lesssim
\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} s\mu\xi_{0} |\partial_{x_1} (\partial_{t}^2-\partial_{x_1}^{2}+\partial_{x_1}^{4}-\partial_{t}\partial_{x_1}^2) \eta |^{2}\ \dt\, \dx_1}
\\
+ \iint_{(0,T)\times\mathcal J_2} e^{2s\varphi_{0}} \left(s^{10}\mu^{11}\xi_{0}^{10}|\eta|^{2}+s^{8}\mu^{9}\xi_{0}^{8}|\partial_{x_1}\eta|^{2} +s^{6}\mu^{7}\xi_{0}^{6}|\partial_{t}\eta|^{2} +s^{2}\mu^{3}\xi_{0}^{2}|\partial_{t}^2\eta|^{2}
\right)\dt\, \dx_1.
\end{multline}
Then proceeding as in~\cite{Sourav}, one can absorb the local terms in $\partial_{x_1}\eta$ and in $\partial_{t}\eta$ by using a cut-off function and integrations by parts and we deduce the result.
\end{proof}

\subsection{A Carleman estimate for the velocity}\label{carvel}

From now on, we take $\mu=\mu_0$ as in \cref{CarBeam} and take $\lambda\geq \mu_0$. The constants that follow in the article may depend on $\mu_0$. We have the following standard Carleman estimate for the heat equation (see, for instance, \cite{FCGBGP} or~\cite{FursikovImanuvilov}). For sake of completeness, we also give a sketch of the proof of the following result in \cref{sec_tecB}.

\begin{theo}\label{CarVel}
Assume $\mu=\mu_0$ and $\omega_0\Subset \omega_1 \Subset \omega$. There exist $s_{0}>0$ and $\lambda_{0}>0$ such that for any $\lambda\geq \lambda_{0}$ $s\geq s_{0}(T^2+T^4)$, and for any smooth function $u$ such that
\[
u=0 \quad \text{on} \ (0,T)\times \Gamma_0, \quad u_1=0 \quad \text{on} \ (0,T)\times \Gamma_1, \quad \frac{\partial u_2}{\partial n}=0 \quad \text{on} \ (0,T)\times \Gamma_1,
\]
we have
\begin{multline}\label{Carlemanonthevelocity}
\iint_{(0,T)\times \Omega} \left(|\nabla^{2}u|^{2}+(\partial_t u)^2 \right) e^{2s\varphi} \ \dt\,\dx\\
 +\iint_{(0,T)\times \Omega} s^2\lambda^{2} \xi^{2} e^{2s\varphi} | \nabla u |^{2} \ \dt\,\dx + \iint_{(0,T)\times \Omega} s^{4}\lambda^{4}\xi^{4} e^{2s\varphi} |u|^{2} \ \dt\,\dx \\
\lesssim
\iint_{(0,T)\times \Omega} s\xi e^{2s\varphi} \left| (\partial_{t}-\Delta) u \right|^{2} \ \dt\,\dx + \iint_{(0,T)\times\omega_1} s^{4}\lambda^{4}\xi^{4}e^{2s\varphi} |u|^{2} \ \dt\,\dx.
\end{multline}
\end{theo}

\subsection{A Carleman estimate for the pressure}\label{carpress}

In order to obtain a Carleman estimate for the pressure, we use that from~\eqref{ns0.3}, the pressure $p_0$ is harmonic in $\Omega$. We recall that $p^\infty$ is defined from $p_0$ by~\eqref{pcp} and~\eqref{utrunc}. In particular, it satisfies the Laplace equation\eqref{ns0.31} but without any explicit boundary condition. Thus in our Carleman estimate, we keep in the right-hand side a boundary term microlocalized in a high frequency regime (represented by $\supp(1-\chi)$, with $\chi$ defined by~\eqref{defchi}). We recall that $\tau$ is defined in~\eqref{deftau}.

\begin{prop}\label{P-LF}
Assume $\mu=\mu_0$ and $\omega_0\Subset \omega_1 \Subset \omega$. There exist $s_{0}>0$ $\lambda_{0}>0$ and $C>0$ such that for any $s\geq s_{0}(T^2+T^4)$, $\lambda\geq \lambda_{0}$ and for any smooth function $p$, the function $p^\infty:=p \chi^\infty$ satisfies
\begin{multline}\label{1918}
\iint_{(0,T)\times \Omega^{\infty}} s^{3}\lambda^{4}\xi^{3} e^{2s\varphi} |p^{\infty}|^{2} \ \dt\,\dx +\iint_{(0,T)\times \Omega^{\infty}} s \lambda^2 \xi e^{2s\varphi} |\nabla p^{\infty} |^{2}\ \dt\,\dx
\\
+\iint_{(0,T)\times \partial\Omega^\infty} s^3 \lambda^3 \xi_0^3 e^{2s\varphi_0} \left| p^{\infty} \right|^{2} \ \dt\, \dx_1 +\iint_{(0,T)\times \partial\Omega^\infty} s \lambda \xi_0 e^{2s\varphi_0} \left| \partial_{x_1} p^{\infty} \right|^{2} \ \dt\, \dx_1
\\
\leq C\left(
\iint_{(0,T)\times \Omega^{\infty}} e^{2s\varphi} |\Delta p^{\infty}|^{2} \ \dt\,\dx +\iint_{(0,T)\times \omega_1} s^{3}\lambda^{4}\xi^{3} e^{2s\varphi} |p^{\infty}|^{2}\ \dt\, \dx
\right.\\ \left. +\iint_{(0,T)\times \partial \Omega^{\infty}} \tau \left| \partial_{x_1}\Op(1-\chi)\left[e^{s\varphi_0} p^{\infty} \right]\right|^2\ \dt\, \dx_1
\right).
\end{multline}
\end{prop}

\begin{proof}
We start by a standard Carleman estimate for $p^\infty$ in $\Omega^\infty$, using that $\chi^\infty$ has a compact support. First, we set
\[
q=e^{s\varphi} p^{\infty}
\]
and we perform standard computations (see, for instance, \cite{FCGBGP, LebeauRobbiano}, \cite[pp.~106--117]{CarlemanBook1}), to obtain the existence of positive constants $c,C,s_0$ such that for $s\geq s_0(T^2+T^4)$,
\begin{multline}\label{1041935}
\!\!\!\!\!\begin{aligned}
&c\iint_{(0,T)\times \Omega^\infty} \left(s^3\lambda^4\xi^3 q^2 +s\lambda^2\xi \left|\nabla q\right|^2 +\frac{1}{s\xi} \left|\Delta q\right|^2 \right) \ \dt\, \dx
\\
&+\iint_{(0,T)\times \partial \Omega^\infty}\!
\left(- s^3\lambda^3\xi^3 \left|\nabla \psi \right|^2 \frac{\partial \psi}{\partial n} q^2 - 2s\lambda^2\xi \left|\nabla \psi \right|^2 \frac{\partial q}{\partial n} q - 2s\lambda\xi \nabla \psi \cdot \nabla q \frac{\partial q}{\partial n} + s\lambda\xi \frac{\partial \psi}{\partial n} \left|\nabla q\right|^2\right) \dt\, \dx_1
\end{aligned}
\\
\leq C\left(\iint_{(0,T)\times \Omega^\infty} \left(-\Delta p\right)^2 e^{2s\varphi}\ \dt \, \dx +\iint_{(0,T)\times \omega_1} s^3\lambda^4\xi^3 q^2 \ \dx\right).
\end{multline}
From~\eqref{defpsi} and~\eqref{rt1055}, we have
\[
\frac{\partial \psi}{\partial x_1}=\frac{\mu_0}{\lambda} \psi_{\mathcal{I}}', \quad
\frac{\partial \psi}{\partial n}=-1 \quad \text{on} \ \partial \Omega^\infty.
\]
Thus there exist $\lambda_0>0$ and $s_0>0$ such that for $\lambda\geq \lambda_0$, $s\geq s_0(T^2+T^4)$, we have on $(0,T)\times \partial\Omega$,
\begin{multline}\label{sa19:19}
- s^3\lambda^3\xi^3 \left|\nabla \psi \right|^2 \frac{\partial \psi}{\partial n} q^2 - 2s\lambda^2\xi \left|\nabla \psi \right|^2 \frac{\partial q}{\partial n} q - 2s\lambda\xi \nabla \psi \cdot \nabla q \frac{\partial q}{\partial n} + s\lambda\xi \frac{\partial \psi}{\partial n} \left|\nabla q\right|^2
\\
=s^3\lambda^3\xi^3 \left|\nabla \psi \right|^2 q^2 + s\lambda\xi \left(\frac{\partial q}{\partial n}\right)^2 - s\lambda\xi \left(\frac{\partial q}{\partial x_1}\right)^2 - 2s\lambda^2\xi \left|\nabla \psi \right|^2 \frac{\partial q}{\partial n} q - 2s\mu_0\xi \psi_{\mathcal{I}}' \frac{\partial q}{\partial x_1} \frac{\partial q}{\partial n}
\\
\geq \frac{1}{2}s^3\lambda^3\xi_0^3 q^2 + \frac{1}{2} s\lambda\xi_0 \left(\frac{\partial q}{\partial n}\right)^2 -2 s\lambda\xi_0 \left(\frac{\partial q}{\partial x_1}\right)^2.
\end{multline}
Let us denote by $\widehat{q}$ the Fourier transform of $q$ in the $x_1$ direction. Then by using the Plancherel theorem, there exists $c>0$ such that
\[
\iint_{(0,T)\times \partial \Omega^\infty} \left(\frac{1}{2}s^3\lambda^3\xi_0^3 q^2 -2 s\lambda\xi_0 \left(\frac{\partial q}{\partial x_1}\right)^2 \right) \ \dx_1\,\dt
\geq c \iint_{(0,T)\times \partial \Omega^\infty} \tau \left(K_-^3 \tau^2 -4 K_+ k^2\right) \left|\widehat{q}\right|^2 \ dk\,\dt
\]
and thus, there exist two constant $c, C>0$ such that
\begin{multline}
\iint_{(0,T)\times \partial \Omega^\infty} \left(\frac{1}{2}s^3\lambda^3\xi_0^3 q^2 -2 s\lambda\xi_0 \left(\frac{\partial q}{\partial x_1}\right)^2 \right) \ \dx_1\,\dt +C
\iint_{(0,T)\times \partial \Omega^\infty} (1-\chi)^2 \tau k^2 \left|\widehat{q}\right|^2 \ dk \,\dt
\\
\geq c\iint_{(0,T)\times \partial \Omega^\infty} \tau \left(\tau^2 + k^2\right) \left|\widehat{q}\right|^2 \ dk \,\dt.
\end{multline}
Using again the Plancherel theorem, and combining the above relation with~\eqref{sa19:19} and with~\eqref{1041935}, we deduce the result.
\end{proof}

\subsection{Gathering the Carleman estimates}\label{cargath}
We are now in a position to prove \cref{P04}
\begin{proof}
Assume that $(u,p_0,\eta)$ is the solution of~\eqref{ns0.3}. We consider $p$ defined from $p_0$ by~\eqref{pcp} and $p^\infty$ defined from $p$ by~\eqref{utrunc}. We apply \cref{C01}, \cref{CarVel} and \cref{P-LF} and using that $\nabla p_0=\nabla p$, we obtain the following relations for $I_1$, $I_2$ and $I_3$ (defined by~\eqref{lu10:26}--\eqref{lu10:28}):
\begin{equation}\label{di16:10}
I_1(s,\lambda,\eta)\lesssim
\iint_{(0,T)\times\mathcal{I}} e^{2s\varphi_{0}} s\lambda \xi_{0} |\partial_{x_1} p |^{2}\ \dt\, \dx_1 +\lambda \iint_{(0,T)\times\mathcal J_1} e^{2s\varphi_{0}} \left(s^{10}\xi_{0}^{10}|\eta|^{2}+s^{2}\xi_{0}^{2}|\partial_{t}^2\eta|^{2}
\right)\ \dt\, \dx_1,
\end{equation}
\begin{equation}\label{di16:11}
I_2(s,\lambda,u)
\lesssim
\iint_{(0,T)\times \Omega} s\lambda^{2} \xi e^{2s\varphi} \left| \nabla p \right|^{2} \ \dt\,\dx + \iint_{(0,T)\times\omega_1} s^{4}\lambda^{6}\xi^{4}e^{2s\varphi} |u|^{2} \ \dt\,\dx,
\end{equation}
and
\begin{multline}\label{di16:12}
I_3(s,\lambda,p^\infty)
\lesssim
\iint_{(0,T)\times \Omega^{\infty}} e^{2s\varphi} |f^{(3)}|^{2} \ \dt\,\dx +\iint_{(0,T)\times \omega_1} s^{3}\lambda^{4}\xi^{3} e^{2s\varphi} |p^{\infty}|^{2}\ \dt\, \dx
\\
+\iint_{(0,T)\times \partial \Omega^{\infty}} \tau \left| \partial_{x_1}\Op(1-\chi)\left[e^{s\varphi_0} p^{\infty} \right]\right|^2\ \dt\, \dx_1.
\end{multline}
Then, we can estimate $f^{(3)}$ by using~\eqref{1519} and we deduce that
\[
\iint_{(0,T)\times \Omega^{\infty}} e^{2s\varphi} |f^{(3)}|^{2} \ \dt\,\dx \leq C \lambda^{-2} I_{3}(s,\lambda,p^{\infty}).
\]
Using that $\chi^\infty\equiv 1$ in $(0,2\pi)\times (0,1)$, and taking $\lambda\geq\lambda_{0}$ with $\lambda_{0}>0$ sufficiently large, we can combine the three Carleman estimates~\eqref{di16:10}--\eqref{di16:12} and the above relation to obtain~\eqref{2322}.
\end{proof}



\section{High frequency estimates}\label{sec_hf}
In this section, we eliminate the last term in~\eqref{2322} by showing high frequency estimates for $u$ and $p$. The method used here is the same as the one used in~\cite{buffe:hal-03331176}. We conjugate the system~\eqref{ns0.31} with $e^{s\varphi_0}$, using that the spatial derivatives of $\varphi_{0}$ involve only powers of $\mu_0$ that is fixed, instead of powers of $\lambda$ for the spatial derivatives of $\varphi$. This allows us to perform energy estimates of the Stokes system, by considering all the terms coming from the conjugaison as lower order terms in the high frequency regime.

\subsection{Estimates from the Stokes system}

We recall that $u^\infty$, $p^\infty$ and $\eta^\infty$ are defined in~\eqref{utrunc} by using the function $\chi^\infty$. We introduce
\begin{equation}\label{09:53}
\widetilde\chi^{\infty}\in C^{\infty}_{0}(\mathbb R,[0,1]),
\quad \widetilde \chi^\infty\equiv 1 \quad \text{in} \ \supp \chi^\infty.
\end{equation}
Then, we set
\begin{align}\label{Checkdefi}
\check u &:=e^{s\varphi_{0}}u^\infty, &\check p &:=e^{s\varphi_{0}}p^\infty, & \check \eta &:=e^{s\varphi_{0}}\eta^\infty,
\\ \label{Widetildedefi}
\widetilde u&:=\Op(1-\chi)\check u, &\widetilde p&:=\Op(1-\chi)\check p, &\widetilde \eta&:=\Op(1-\chi)\check \eta,
\\ \label{Widetildeinftydefi2}
\widetilde u^{\infty}&:=\widetilde \chi^{\infty}\Op(1-\chi) \check u,
&
\widetilde p^{\infty}&:=\widetilde \chi^{\infty}\Op(1-\chi)\check p,
&
\widetilde \eta^{\infty}&:=\widetilde \chi^{\infty}\Op(1-\chi) \check \eta.
\end{align}
Our aim is to estimate $\widetilde p$ (see~\eqref{2322}) but we need to use $\widetilde \chi^{\infty}$ to work on a bounded domain and to apply the elliptic regularity of the Stokes system. In order to estimate $\widetilde p$, we use that, with our choice of truncation functions, we have the relations
\[
\widetilde p=\widetilde p^{\infty}+ [1-\widetilde \chi^{\infty},\Op(1-\chi)]\check p.
\]
Then, using the commutator property in \cref{Commutatorcorollary}, we can estimate $\widetilde p$ from $\widetilde p^\infty$ and $\check p$.

Using~\eqref{sa19:07}, \eqref{1131123} and~\eqref{1131123bis}, we have
\[
\tau \lesssim s\lambda \xi_0, \quad \tau \lesssim s\lambda \xi.
\]
This leads us to define (see~\eqref{lu10:26}--\eqref{lu10:28})
\begin{multline}\label{1812}
I_4(s,\lambda,\check \eta):=
\iint_{(0,T)\times\mathcal{I}} \left(
\lambda^{-9} \tau^{10} |\check \eta |^{2} +\lambda^{-7} \tau^{8} |\partial_{x_1}\check \eta |^{2} +\lambda^{-5} \tau^{6} \left(| \partial_{x_1}^2 \check\eta |^{2} +| \partial_{t} \check \eta |^{2} \right)
\right)
\ \dt\, \dx_1
\\
\begin{aligned}
&+\iint_{(0,T)\times\mathcal{I}} \left(
\lambda^{-3}\tau^{4} \left(| \partial_{x_1}^{3}\check \eta |^{2}+ |\partial_{x_1} \partial_{t}\check \eta |^{2}\right)
\right)
\ \dt\, \dx_1
\\
&+\iint_{(0,T)\times\mathcal{I}} \left(
\lambda^{-1}\tau^2 \biggl(| \partial_{x_1}^{4}\check \eta |^{2}+ |\partial_{t}\partial_{x_1}^2\check \eta |^{2}+ |\partial_{t}^2\check \eta |^{2}\right)
\end{aligned} \\
+ \lambda
\left(\left| \partial_{x_1}^{5}\check \eta \right|^{2}+ | \partial_{t}^2\partial_{x_1}\check \eta |^{2}+ | \partial_{t}\partial_{x_1}^3 \check \eta |^{2}\right)\biggr)\ \dt\, \dx_1,
\end{multline}
\begin{equation}\label{1813}
I_5(s,\lambda,\check u):=
\lambda^{2} \iint_{(0,T)\times \Omega} \left(|\nabla^{2}\check u|^{2}+(\partial_t \check u)^2 +\tau^{2} | \nabla \check u |^{2} +\tau^{4} | \check u |^{2}
\right) \ \dt\,\dx,
\end{equation}
and
\begin{equation}\label{1814}
I_6(s,\lambda,\check p):=
\lambda \iint_{(0,T)\times \Omega^{\infty}} \left(\tau^{3} |\check p|^{2}+\tau |\nabla \check p|^{2} \right)\ \dt\,\dx +\iint_{(0,T)\times \partial\Omega^\infty} \left(\tau^3 \left|\check p \right|^{2}+\tau \left|\partial_{x_1} \check p \right|^{2} \right) \ \dt\, \dx_1.
\end{equation}
Noting that
\[
I_{4}(s,\lambda,\check\eta)\lesssim I_{1}(s,\lambda,\eta), \quad I_{5}(s,\lambda,\check u) \lesssim I_{2}(s,\lambda,u), \quad
I_{6}(s,\lambda,\check p) \lesssim I_{3}(s,\lambda,p^{\infty}),
\]
we deduce from~\eqref{2322} that
\begin{multline}\label{2323}
\!\! I_4(s,\lambda,\check \eta)+I_5(s,\lambda,\check u)+I_6(s,\lambda,\check p)
\lesssim
\lambda \iint_{(0,T)\times\mathcal J_1}\!\! e^{2s\varphi_{0}} \left(s^{10}\xi_{0}^{10}|\eta|^{2}+s^{2}\xi_{0}^{2}|\partial_{t}^2\eta|^{2}
\right) \dt\, \dx_1
\\
\begin{aligned}[b]
&+ \iint_{(0,T)\times\omega_1} e^{2s\varphi}\left(s^{4}\lambda^{6}\xi^{4} |u|^{2} \ \dt\,\dx +s^{3}\lambda^{4}\xi^{3} |p^{\infty}|^{2}\right)\ \dt\, \dx\\
& +\iint_{(0,T)\times \partial \Omega^{\infty}} \tau \left| \partial_{x_1}\widetilde{p}\right|^2\ \dt\, \dx_1.
\end{aligned}\!\!
\end{multline}
The aim of this section is to show the following result:

\begin{prop}\label{P05}
There exist $\lambda_0>0$ and $s_0>0$ such that for $\lambda\geq \lambda_0$ and $s\geq s_0(T^2+T^4)$, any smooth solutions $[u,p_0,\eta]$ of~\eqref{ns0.2} satisfies
\begin{multline}\label{1014}
I_4(s,\lambda,\check \eta)+I_5(s,\lambda,\check u)
\lesssim
\lambda \iint_{(0,T)\times\mathcal J_1} e^{2s\varphi_{0}} \left(s^{10}\xi_{0}^{10}|\eta|^{2}+s^{2}\xi_{0}^{2}|\partial_{t}^2\eta|^{2}
\right)\ \dt\, \dx_1
\\
+ \iint_{(0,T)\times\omega_1} e^{2s\varphi}\left(s^{4}\lambda^{6}\xi^{4} |u|^{2} \ \dt\,\dx +s^{3}\lambda^{4}\xi^{3} |p|^{2}
\right)\ \dt\, \dx,
\end{multline}
where $\check \eta$ and $\check u$ are defined by~\eqref{Checkdefi} and $p$ is defined by~\eqref{pcp}.
\end{prop}

Before proving \cref{P05}, let us first introduce some preliminary results and notation. Recalling that $\chi$ is defined in~\eqref{defchi}, we deduce that if $\chi\neq 1$ then
\begin{equation}\label{hfk}
\tau \lesssim |k|.
\end{equation}
This yields the following semi-classical trace inequality:

\begin{lemm}\label{Traceformulalemma}
There exists $s_{0}>0$ such that for any $s\geq s_0 T^4$ and for any $f\in H^1(\Omega^\infty)$,
\[
\tau^{1/2} \left\|\Op(1-\chi) f_{|_{\partial\Omega^{\infty}}} \right\|_{L^2(\partial\Omega^\infty)} \lesssim \left\| \nabla \Op(1-\chi) f \right\|_{L^2(\Omega^\infty)}.
\]
\end{lemm}
\begin{proof}
We write $g:=\Op(1-\chi) f$ and
\[
g^2(x_1,1)=g^2(x_1,x_2)+ 2 \int_{x_2}^1 g(x_1,y_2) \partial_{x_2}g (x_1,y_2) \ dy_2
\]
so that
\[
\tau \int_{\Gamma_1^\infty} g(x_{1},1)^2 \ \dx_1 \leq \tau \left\| g\right\|_{L^2(\Omega^\infty)}^2 + 2\tau \left\| g\right\|_{L^2(\Omega^\infty)}\left\| \partial_{x_2} g\right\|_{L^2(\Omega^\infty)}
\leq (\tau+4\tau^2) \left\| g\right\|_{L^2(\Omega^\infty)}^2 + \left\| \partial_{x_2} g\right\|_{L^2(\Omega^\infty)}^2
\]
and we conclude by using~\eqref{hfk}.
\end{proof}

In particular, using \cref{Traceformulalemma}, we can estimate the last term of~\eqref{2323} as follows:
\begin{equation}\label{1546}
\iint_{(0,T)\times \partial \Omega^{\infty}} \tau \left| \partial_{x_1} \widetilde{p}_{|_{\partial\Omega^{\infty}}} \right|^2\ \dt\, \dx_1
\lesssim \left\| \nabla \partial_{x_1} \widetilde p\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2.
\end{equation}

We have also the following result that will allows us to estimate boundary terms:

\begin{lemm}\label{L03}
There exists $s_{0}>0$ such that for any $s\geq s_0 T^4$ and for any $f\in H^2(\mathbb{R})$,
\[
\left\|\Op(1-\chi) f \right\|_{H^{3/2}(\mathbb{R})} \lesssim
\tau^{-1/2} \left\|\Op(1-\chi) \partial_{x_1}^2 f \right\|_{L^{2}(\mathbb{R})}.
\]
\end{lemm}
\begin{proof}
Denoting by $\widehat f$ the Fourier transform of $f$, we have
\[
\left\|\Op(1-\chi) f \right\|_{H^{3/2}(\mathbb{R})}^2=\int_{\mathbb{R}} (1+k^2)^{3/2} (1-\chi(\tau,k))^2 \left| \widehat{f}(k)\right|^2 \ dk
\]
and by using~\eqref{hfk}, we deduce the result.
\end{proof}

In order to estimate $\widetilde{p}$ (and prove \cref{P05}), we consider the system verified by $\widetilde{u}^{\infty}$ and $\widetilde{p}^{\infty}$: from~\eqref{ns0.31}, we have
\begin{equation}\label{ns0.3tilde}
\begin{cases}
\partial_{t} \widetilde{u}^{\infty} - \Delta \widetilde{u}^{\infty} + \nabla \widetilde{p}^{\infty} = \widetilde f^{(1)}& \text{in} \ (0,T)\times \Omega^\infty,\\
\div \widetilde{u}^{\infty}= \widetilde f^{(2)}& \text{in} \ (0,T)\times \Omega^\infty,\\
\widetilde{u}^{\infty}=0 & \text{on} \ (0,T)\times\Gamma_0^\infty,\\
\widetilde{u}^{\infty}=\widetilde h e_2 & \text{on} \ (0,T)\times \Gamma_1^\infty,\\
\widetilde{u}^{\infty}(0,\cdot\,)=\widetilde u^{\infty} (T,\cdot\,)=0 & \text{in} \ \Omega^\infty,
\end{cases}
\end{equation}
where
\begin{multline}\label{deff}
\!\!\!\!\!\widetilde f^{(1)} =\widetilde \chi^{\infty} \Op(1-\chi) e^{s\varphi_{0}} f^{(1)} + (s\partial_{t}\varphi_{0})\widetilde{u}^{\infty} -s\left(\partial^{2}_{x_{1}}\varphi_0\right) \widetilde{u}^{\infty} +s^{2}\left(\partial_{x_{1}}\varphi_{0}\right)^{2}\widetilde{u}^{\infty} -2s\partial_{x_{1}}\varphi_{0}\partial_{x_{1}}\widetilde{u}^{\infty} +s\nabla\varphi_{0}\widetilde{p}^{\infty}
\\
\begin{aligned}[b]
&+\left[-(s\partial_{t}\varphi_{0})+s\left(\partial^{2}_{x_{1}}\varphi_0\right)-s^{2}\left(\partial_{x_{1}}\varphi_{0}\right)^{2} +2s\partial_{x_{1}}\varphi_{0}\partial_{x_{1}},\widetilde\chi^{\infty}\Op(1-\chi)\right]\check u
\\
&+[-s\nabla \varphi_0,\widetilde\chi^{\infty}\Op(1-\chi)]\check p -\widetilde\chi^{\infty}\Op(\partial_t \chi)\check{u} -\left(\widetilde\chi^{\infty}\right)''\widetilde{u} -2 \left(\widetilde\chi^{\infty}\right)'\partial_{x_1} \widetilde{u} +\left(\widetilde\chi^{\infty}\right)'\widetilde{p}e_1,
\end{aligned}
\end{multline}
\begin{equation}\label{defg}
\widetilde f^{(2)}=\widetilde \chi^{\infty}\Op(1-\chi)(e^{s\varphi_{0}}f^{(2)})+s\partial_{x_1} \varphi_{0} \widetilde{u}_1^{\infty} -[s\partial_{x_1} \varphi_{0},\widetilde\chi^{\infty}\Op(1-\chi)] \check u_1+\left(\widetilde\chi^{\infty}\right)' \widetilde{u}_1,
\end{equation}
\begin{equation}\label{defh}
\widetilde h:=\widetilde\chi^{\infty}\Op(1-\chi)(\partial_{t}\check\eta-s\left(\partial_t \varphi_0\right)\check \eta).
\end{equation}
We also define
\begin{multline}\label{apriori3}
\widetilde{I}(\widetilde u, \widetilde p):=
\left\| \partial_t \partial_{x_1}\widetilde u\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\left\| \partial_{x_1} \widetilde u\right\|_{L^2(0,T;H^2(\Omega^\infty))}^2 + \left\| \nabla \partial_{x_1} \widetilde p\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
\\
\begin{aligned}
&+\left\| \tau \partial_t \widetilde u\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\left\| \tau^3 \widetilde u\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\left\| \tau^2 \partial_{x_1}\widetilde u\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
\\
&+\left\| \tau \partial_{x_1}^2 \widetilde u\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
%\\
+\left\| \tau \partial_{x_1}\partial_{x_2} \widetilde u\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\left\| \tau \partial_{x_2}^2 \widetilde u\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
\end{aligned}\\
+\left\| \tau^2 \partial_{x_2} \widetilde u\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
%\\
+ \left\| \tau^2 \widetilde p\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 + \left\| \tau \nabla \widetilde p\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2.
\end{multline}
From relation~\eqref{hfk}, we have
\begin{equation}\label{11:03}
\widetilde{I}(\widetilde u, \widetilde p)\lesssim \left\| \partial_t \partial_{x_1}\widetilde u\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\left\| \partial_{x_1} \widetilde u\right\|_{L^2(0,T;H^2(\Omega^\infty))}^2 + \left\| \nabla \partial_{x_1} \widetilde p\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2.
\end{equation}

We have the following a priori estimate on~\eqref{ns0.3tilde}.

\begin{prop}
There exist $\lambda_0>0$ and $s_0>0$ such that for $\lambda\geq \lambda_0$ and $s\geq s_0(T^2+T^4)$, any smooth solutions $[u,p_0,\eta]$ of~\eqref{ns0.2} satisfies
\begin{multline}\label{apriori2}
\!\!\!\widetilde{I}(\widetilde u, \widetilde p)\lesssim
\left\| \partial_{x_1} \widetilde f^{(1)}\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\left\| \partial_{x_1} \widetilde f^{(2)}\right\|_{L^2(0,T;H^1(\Omega^\infty))}^2 +\left\| \partial_{t} \widetilde f^{(2)}\right\|^{2}_{L^{2}(0,T;L^{2}(\Omega^{\infty}))}
\\
+\left\| \partial_{x_1} \widetilde h \right\|_{L^2(0,T;H^{3/2}(\Gamma_1^\infty))}^2 + \left\| \tau^{-1/2} \partial_t \partial_{x_1} \widetilde{h} \right\|_{L^2(0,T;L^2(\Gamma_1^\infty))}^2 +\lambda^{-1} \left(I_{5}(s,\lambda,\check u)+I_{6}(s,\lambda,\check p)\right).\!\!\!
\end{multline}
\end{prop}

\begin{proof}
First, we differentiate~\eqref{ns0.3tilde} with respect to $x_1$:
\begin{equation}\label{di19:57}
\begin{cases}
\partial_{t} \partial_{x_1}\widetilde{u}^{\infty} - \Delta \partial_{x_1} \widetilde{u}^{\infty} + \nabla \partial_{x_1} \widetilde{p}^{\infty} = \partial_{x_1} \widetilde f^{(1)}& \text{in} \ (0,T)\times \Omega^\infty,\\
\div \partial_{x_1} \widetilde{u}^{\infty}= \partial_{x_1} \widetilde f^{(2)}& \text{in} \ (0,T)\times \Omega^\infty,\\
\partial_{x_1} \widetilde{u}^{\infty}=0 & \text{on} \ (0,T)\times\Gamma_0^\infty,\\
\partial_{x_1} \widetilde{u}^{\infty}=\partial_{x_1} \widetilde h e_2 & \text{on} \ (0,T)\times \Gamma_1^\infty,\\
\partial_{x_1} \widetilde{u}^{\infty}(0,\cdot\,)=\partial_{x_1} \widetilde u^{\infty} (T,\cdot\,)=0 & \text{in} \ \Omega^\infty.
\end{cases}
\end{equation}
Let us consider a bounded smooth domain $\Omega^{\natural}\subset \Omega^\infty$ containing $\overline{\supp \widetilde\chi^{\infty}}\times (0,1)$. Let us also write
\[
h^{\natural}=
\begin{cases}
\partial_{x_1} \widetilde{u}^{\infty}=0 & \text{on} \ (0,T)\times\left(\partial\Omega^{\natural} \setminus \Gamma_1^\infty\right),\\
\partial_{x_1} \widetilde{u}^{\infty}=\partial_{x_1} \widetilde h e_2 & \text{on} \ (0,T)\times \left(\partial\Omega^{\natural} \cap \Gamma_1^\infty\right).
\end{cases}
\]
Using~\cite[p.~33, Theorem~7.5]{LionsMagenes1}, there exists $H\in H^2(\left\{ (x_1,x_2)\in \mathbb{R}^2 \ ; \ x_2<1 \right\})$ such that $H=\partial_{x_1} \widetilde h$ on $\Gamma_1^\infty$. Multiplying $H$ by an adequate cut-off function we deduce the existence of $H^{\natural}\in H^2(\Omega^{\natural})$ such that $H^\natural=h^\natural$ on $\partial\Omega^\natural$. Therefore $h^\natural\in H^{3/2}(\partial\Omega^\natural)$ and we have the estimate
\[
\left\| h^\natural \right\|_{ H^{3/2}(\partial\Omega^\natural)} \lesssim \left\| \partial_{x_1} \widetilde h \right\|_{H^{3/2}(\Gamma_1^\infty)}.
\]
With the above notation, we deduce from~\eqref{di19:57} that $\left(\partial_{x_1} \widetilde{u}^{\infty}, \partial_{x_1} \widetilde{p}^{\infty}\right)$ satisfies a Stokes system in $\Omega^{\natural}$:
\begin{equation}\label{di19:57-bis}
\begin{cases}
- \Delta \partial_{x_1} \widetilde{u}^{\infty} + \nabla \partial_{x_1} \widetilde{p}^{\infty} = \partial_{x_1} \widetilde f^{(1)} -\partial_{t} \partial_{x_1}\widetilde{u}^{\infty}& \text{in} \ (0,T)\times \Omega^{\natural},\\
\div \partial_{x_1} \widetilde{u}^{\infty}= \partial_{x_1} \widetilde f^{(2)}& \text{in} \ (0,T)\times \Omega^{\natural},\\
\partial_{x_1} \widetilde{u}^{\infty}=h^\natural & \text{on} \ (0,T)\times \partial\Omega^{\natural}.
\end{cases}
\end{equation}
Using the elliptic regularity of the Stokes system (see, for instance, \cite[Proposition~2.2 p.~33]{Temam}) we obtain
\begin{multline}\label{1018}
\left\| \partial_{x_1}\widetilde u^{\infty}\right\|_{L^2(0,T;H^2(\Omega^\infty))}^2 + \left\| \nabla \partial_{x_1} \widetilde p^{\infty}\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2\\
\lesssim
\left\| \partial_{x_1} \widetilde f^{(1)}\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
+\left\| \partial_t \partial_{x_1} \widetilde{u}^{\infty} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2\\
+\left\| \partial_{x_1} \widetilde f^{(2)}\right\|_{L^2(0,T;H^1(\Omega^\infty))}^2 +\left\| \partial_{x_1} \widetilde h \right\|_{L^2(0,T;H^{3/2}(\Gamma_1^\infty))}^2.
\end{multline}
On the other hand, by multiplying the first equation of~\eqref{di19:57} by $\partial_t \partial_{x_1} \widetilde{u}$ and integrating by parts, we deduce
\begin{multline*}
\int_0^T \left\| \partial_t \partial_{x_1} \widetilde u^{\infty}\right\|_{L^2(\Omega^\infty)}^2 \ \dt +\iint_{(0,T)\times \Gamma_1^\infty} \partial_{x_1} \widetilde{p}^{\infty}_{|_{\Gamma_1^\infty}} \overline{\partial_t \partial_{x_1}\widetilde{h}} \ \dx_1\dt +\iint_{(0,T)\times \Omega^\infty} \partial_{x_{1}}^2 \widetilde p^{\infty} \overline{\partial_{t} \widetilde f^{(2)}} \ \dt \dx
\\
=\iint_{(0,T)\times \Omega^\infty} \partial_{x_1} \widetilde{f}^{(1)}\cdot \overline{\partial_t \partial_{x_1} \widetilde u^{\infty}} \ \dx\dt.
\end{multline*}
The above relation yields that for any $\varepsilon >0$,
\begin{multline*}
\left\| \partial_t \partial_{x_1} \widetilde u^{\infty}\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2\\
\lesssim \varepsilon \left\| \nabla \partial_{x_1} \widetilde{p}^{\infty}\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
+\varepsilon \left\| \tau^{1/2} \partial_{x_{1}}\widetilde p^{\infty} \right\|_{L^{2}(0,T;L^2(\Gamma_1^\infty))}^2
+\frac{1}{\eps} \left\| \partial_{t} \widetilde f^{(2)}\right\|^{2}_{L^{2}(0,T;L^{2}(\Omega^{\infty}))}
\\
+ \frac{1}{\varepsilon} \left\| \tau^{-1/2} \partial_t \partial_{x_1} \widetilde{h} \right\|_{L^2(0,T;L^2(\Gamma_1^\infty))}^2 +\left\| \partial_{x_1} \widetilde{f}^{(1)}\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2.
\end{multline*}
We deduce from the above relation, from~\eqref{Widetildeinftydefi2} and from \cref{Traceformulalemma} that
\begin{multline}\label{11:46}
\left\| \partial_t \partial_{x_1} \widetilde u^{\infty}\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2\\
\lesssim \varepsilon \left\| \nabla \partial_{x_1} \widetilde{p}\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\varepsilon \left\| \widetilde{p}\right\|_{L^2(0,T;H^1(\Omega^\infty))}^2
+\frac{1}{\eps} \left\| \partial_{t} \widetilde f^{(2)}\right\|^{2}_{L^{2}(0,T;L^{2}(\Omega^{\infty}))}\\
 + \frac{1}{\varepsilon} \left\| \tau^{-1/2} \partial_t \partial_{x_1} \widetilde{h} \right\|_{L^2(0,T;L^2(\Gamma_1^\infty))}^2 +\left\| \partial_{x_1} \widetilde{f}^{(1)}\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2.
\end{multline}

Now using~\eqref{09:53} and~\eqref{utrunc}, we have
\[
\nabla^{2} \partial_{x_{1}}\widetilde u = \nabla^{2} \partial_{x_{1}} \widetilde u^{\infty} + \nabla^{2} \partial_{x_{1}} \left[(1-\widetilde \chi^{\infty}),\Op(1-\chi)\right] \check u.
\]
Since, $(1-\widetilde \chi^{\infty}), 1-\chi\in \Sbf^{0}_{\tau}$, we deduce from \cref{Commutatorcorollary} and \cref{Continuitytheorem} that
\begin{multline*}
\left\| \nabla^{2} \partial_{x_{1}}\widetilde u \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
\lesssim \left\| \nabla^{2} \partial_{x_{1}}\widetilde u^{\infty} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 + \left\|\tau^2 \check u\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
\\
+ \left\| \tau \nabla \check u\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 + \left\| \nabla^2 \check u\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2.
\end{multline*}
Using the compact support of $\chi^\infty$ and the periodicity of $u e^{s\varphi_0}$, we deduce from the above relation and from~\eqref{1813} that
\begin{equation}\label{10:19}
\left\| \nabla^{2} \partial_{x_{1}}\widetilde u \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
\lesssim \left\| \nabla^{2} \partial_{x_{1}}\widetilde u^{\infty} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\lambda^{-2} I_{5}(s,\lambda,\check u).
\end{equation}
Similarly, with~\eqref{1814}, we have
\begin{equation}\label{10:29}
\left\| \nabla \partial_{x_{1}}\widetilde p \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
\lesssim \left\| \nabla \partial_{x_{1}}\widetilde p^{\infty} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\lambda^{-1} I_{6}(s,\lambda,\check p).
\end{equation}
Finally,
\[
\partial_{t}\partial_{x_{1}} \widetilde u = \partial_{t}\partial_{x_{1}} \widetilde u^{\infty} - \partial_{x_{1}} [1-\widetilde \chi^{\infty},\Op(\partial_{t}\chi)]\check u +\partial_{x_{1}} [1-\widetilde \chi^{\infty},\Op(1-\chi)] \partial_{t} \check u,
\]
and from \cref{L02},
\[
\partial_{t}\chi \in \frac{\ell'}{(\lambda s)^{1/2} }\tau^{\frac{1}{2}}\Sbf^{0}_{\tau}.
\]
In particular, if $s\geq T^2$, we can combine the two previous relations with \cref{Commutatorcorollary} and \cref{Continuitytheorem} to obtain
\begin{equation}\label{11:35}
\left\| \partial_{t} \partial_{x_{1}}\widetilde u \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
\lesssim \left\| \partial_{t} \partial_{x_{1}}\widetilde u^{\infty} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\lambda^{-2} I_{5}(s,\lambda,\check u).
\end{equation}
Combining~\eqref{11:03}, \eqref{1018}, \eqref{11:46}, \eqref{10:19}, \eqref{10:29} and~\eqref{11:35}, we deduce the result by taking $\varepsilon>0$ small enough.
\end{proof}

Combining~\eqref{apriori2}, \eqref{1546} and~\eqref{2323}, we deduce the existence of $\lambda_0>0$ and $s_0>0$ such that for $\lambda\geq \lambda_0$ and $s\geq s_0(T^2+T^4)$,
\begin{multline}\label{1619}
I_4(s,\lambda,\check \eta)+I_5(s,\lambda,\check u)+I_6(s,\lambda,\check p)+\widetilde{I}(\widetilde u, \widetilde p)\\
\begin{aligned}[b]
&\lesssim
\lambda \iint_{(0,T)\times\mathcal J_1} e^{2s\varphi_{0}} \left(s^{10}\xi_{0}^{10}|\eta|^{2}+s^{2}\xi_{0}^{2}|\partial_{t}^2\eta|^{2}
\right)\ \dt\, \dx_1
\\
&\qquad+ \iint_{(0,T)\times\omega_1} e^{2s\varphi}\left(s^{4}\lambda^{6}\xi^{4} |u|^{2} \ \dt\,\dx +s^{3}\lambda^{4}\xi^{3} |p^{\infty}|^{2}
\right)\ \dt\, \dx
\\
&\qquad+\left\| \partial_{x_1}\widetilde{f}^{(1)}\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\left\|\partial_{x_1} \widetilde f^{(2)}\right\|_{L^2(0,T;H^1(\Omega^\infty))}^2 +\left\|\partial_{t}\widetilde{f}^{(2)}\right\|^{2}_{L^{2}(0,T;L^{2}(\Omega^{\infty}))}
\\
&\qquad+\left\| \tau^{-1/2} \partial_t \partial_{x_1}\widetilde{h} \right\|_{L^2(0,T;L^2(\Gamma_1^\infty))}^2 +\left\| \partial_{x_1} \widetilde h \right\|_{L^2(0,T;H^{3/2}(\Gamma_1^\infty))}^2.
\end{aligned}
\end{multline}

\subsection{Estimates of \texorpdfstring{$\widetilde f^{(1)}$}{f(1)}, \texorpdfstring{$\widetilde f^{(2)}$}{f(2)} and \texorpdfstring{$\widetilde h$}{h}}

To obtain \cref{P05}, it remains to estimate the right-hand side of~\eqref{1619}. We recall that $\widetilde f^{(1)}$, $\widetilde f^{(2)}$ and $\widetilde h$ are given by~\eqref{deff}, \eqref{defg} and~\eqref{defh}.

Combining~\eqref{1700}, \eqref{sa19:07}, \eqref{1131123bis} and~\eqref{Smtau}, we deduce that for $\alpha\geq 0$,
\begin{equation}\label{1702}
\left| \partial_{x_1}^\alpha \varphi_0 \right| \lesssim \frac{\tau}{\lambda s} \quad (k\geq 1), \quad
\left| \partial_t \partial_{x_1}^{\alpha} \varphi_0 \right| \lesssim T\left(\frac{\tau}{\lambda s}\right)^{3/2},\quad
\left| \partial_t^2 \partial_{x_1}^{\alpha} \varphi_0 \right| \lesssim T^2\left(\frac{\tau}{\lambda s}\right)^{2}
\end{equation}
\begin{multline}\label{lu10:44}
\partial_{x_1}^{\alpha} \varphi_0 \in \frac{\tau}{\lambda s} \Sbf^{0}_{\tau} \quad (k\geq 1),\quad
\partial_t \partial_{x_1}^{\alpha} \varphi_0 \in \ell' \left(\frac{\tau}{\lambda s}\right)^{3/2}
(e^{-4\lambda\Psi}+ e^{-2\lambda\Psi})
\Sbf^{0}_{\tau}, \\
\partial_t^2 \partial_{x_1}^{\alpha} \varphi_0 \in \left(2\ell+3(\ell')^2\right)\left(\frac{\tau}{\lambda s}\right)^{2}
(e^{-8\lambda\Psi}+e^{-6\lambda\Psi})
\Sbf^{0}_{\tau}.
\end{multline}

\begin{prop}\label{P01}
There exist $s_0>0$ and $\lambda_0>0$ such that the function $\widetilde f^{(1)}$ defined by~\eqref{deff} satisfies for $s\geq s_0(T^2+T^4)$ and for $\lambda\geq \lambda_0$,
\[
\left\| \partial_{x_1}\widetilde{f}^{(1)}\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
\leq \lambda^{-1} \left(I_4(s,\lambda,\check \eta)+I_5(s,\lambda,\check u)+I_6(s,\lambda,\check p)+\widetilde{I}(\widetilde u, \widetilde p)
\right).
\]
\end{prop}

\begin{proof}
Differentiating~\eqref{deff} yields,
\begin{equation}\label{ddeff}
\partial_{x_1} \widetilde f^{(1)} =\sum_{i=1}^6 F^{(i)}
\end{equation}
where
\begin{align}\label{ddeff1}
F^{(1)}&:=\partial_{x_1} \left(\widetilde\chi^{\infty } \Op(1-\chi) e^{s\varphi_{0}} f^{(1)}\right),
\\\label{ddeff2}
F^{(2)}&:=\partial_{x_1} \left((s\partial_{t}\varphi_{0})\widetilde{u}^{\infty} -s\left(\partial^{2}_{x_{1}}\varphi_0\right) \widetilde{u}^{\infty} +s^{2}\left(\partial_{x_{1}}\varphi_{0}\right)^{2}\widetilde{u}^{\infty} -2s\partial_{x_{1}}\varphi_{0}\partial_{x_{1}}\widetilde{u}^{\infty} +s\nabla\varphi_{0}\widetilde{p}^{\infty}
\right),
\\\label{ddeff3}
F^{(3)}&:=\partial_{x_1} \left[-(s\partial_{t}\varphi_{0})+s\left(\partial^{2}_{x_{1}}\varphi_0\right)-s^{2}\left(\partial_{x_{1}}\varphi_{0}\right)^{2} +2s\partial_{x_{1}}\varphi_{0}\partial_{x_{1}},\widetilde \chi^{\infty}\Op(1-\chi)\right]\check u,
\\\label{ddeff45}
F^{(4)}&:=\partial_{x_1} [-s\nabla \varphi_0,\widetilde \chi^{\infty}\Op(1-\chi)]\check p,\quad F^{(5)}:=-\partial_{x_1} \left(\widetilde \chi^{\infty}\Op(\partial_t \chi)\check{u}\right),
\\\label{ddeff6}
F^{(6)}&:=\partial_{x_1} \left(-\left(\widetilde\chi^{\infty}\right)''\widetilde{u} -2 \left(\widetilde\chi^{\infty}\right)'\partial_{x_1} \widetilde{u} +\left(\widetilde\chi^{\infty}\right)'\widetilde{p}e_1
\right).
\end{align}
From~\eqref{ddeff1}, we have
\begin{equation*}
F^{(1)}= \widetilde\chi^{\infty}\Op(1-\chi)\left(s\partial_{x_1}\varphi_0 e^{s\varphi_{0}} f^{(1)}+e^{s\varphi_{0}} \partial_{x_1}f^{(1)}\right) +(\widetilde\chi^{\infty})'\Op(1-\chi)e^{s\varphi_{0}}f^{(1)}.
\end{equation*}
From~\eqref{1519}, \eqref{1702}, the properties of $\chi^\infty$ and the periodicity of $u$ and $p$ in the $x_1$ variable,
\begin{multline*}
\left\| s\partial_{x_1}\varphi_0 e^{s\varphi_{0}} f^{(1)}+e^{s\varphi_{0}} \partial_{x_1}f^{(1)}\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\left\| e^{s\varphi_{0}}f^{(1)} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 \\
\lesssim \iint_{(0,T)\times (0,2\pi) \times (0,1)} e^{2s\varphi_0} \left(
\frac{\tau^2}{\lambda^2} \left(\left|u\right|^2+\left|\partial_{x_1} u\right|^2 +\left|p\right|^2\right) + \left|\partial_{x_1}^2 u\right|^2+\left|\partial_{x_1} p\right|^2
\right)\ \dx\,\dt
\\
\lesssim \iint_{(0,T)\times \Omega^\infty}
\left(
\left| \partial_{x_1}^2 \check u \right|^2 + \left(\frac{\tau}{\lambda}\right)^2 \left| \partial_{x_1} \check u\right|^2 + \left(\frac{\tau}{\lambda}\right)^4 \left| \check u \right|^2 +\left| \partial_{x_1} \check p \right|^2 + \left(\frac{\tau}{\lambda}\right)^2\left| \check p\right|^2
\right)\ \dx\,\dt.
\end{multline*}
Since $1-\chi\in\Sbf^{0}_{\tau}$ (see \cref{L01}), we deduce from the above estimate, from \cref{Continuitytheorem}, and from~\eqref{1813}-\eqref{1814}, that
\begin{equation}\label{1206}
\left\| F^{(1)}\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
\leq \lambda^{-1} \left(I_5(s,\lambda,\check u)+I_6(s,\lambda,\check p)\right).
\end{equation}
From~\eqref{lu11:34}, \eqref{ddeff2} and~\eqref{1702}, we have for $s\geq s_0 \left(T^2+T^4\right)$ and $\lambda\geq \lambda_0$,
\begin{equation}\label{1104}
\left|F^{(2)} \right|
\lesssim \frac{\tau^{2}}{\lambda^{2}} \left(\left| \widetilde{u}\right| + \left|\partial_{x_1} \widetilde{u}\right| \right) +\frac{\tau}{\lambda} \left(\left|\partial_{x_1}^2 \widetilde{u}\right|+\left|\widetilde{p}\right|+\left|\partial_{x_1} \widetilde{p}\right| \right),
\end{equation}
and thus with~\eqref{apriori3},
\begin{equation}\label{1839}
\left\| F^{(2)}\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
\leq \lambda^{-2} \widetilde{I}(\widetilde u, \widetilde p).
\end{equation}
On the other hand, from~\eqref{ddeff3}, \eqref{ddeff45},
\begin{multline}\label{1330}
F^{(3)} = \left[-(s\partial_{t}\varphi_{0})+s\left(\partial^{2}_{x_{1}}\varphi_0\right)-s^{2}\left(\partial_{x_{1}}\varphi_{0}\right)^{2} +2s\partial_{x_{1}}\varphi_{0}\partial_{x_{1}},\widetilde\chi^{\infty} \Op(1-\chi)\right]\partial_{x_1} \check{u}
\\
+\left[-(s\partial_{t}\partial_{x_1}\varphi_{0})+s\left(\partial^{3}_{x_{1}}\varphi_0\right)-2s^{2}\partial_{x_{1}}\varphi_{0}\partial_{x_{1}}^2\varphi_{0} +2s\partial_{x_{1}}^2\varphi_{0}\partial_{x_{1}} +2s\partial_{x_{1}}\varphi_{0}\partial_{x_{1}}^2,\widetilde\chi^{\infty} \Op(1-\chi)\right]\check u,
\end{multline}
\begin{equation}\label{1331}
F^{(4)}= [-s\nabla \varphi_0,\widetilde\chi^{\infty} \Op(1-\chi)]\partial_{x_1}\check{p}+ [-s\nabla \partial_{x_1} \varphi_0,\widetilde\chi^{\infty} \Op(1-\chi)]\check p.
\end{equation}
From~\eqref{lu10:44},
\[
s\left(\partial^{2}_{x_{1}}\varphi_0\right)-s^{2}\left(\partial_{x_{1}}\varphi_{0}\right)^{2} -2s\partial_{x_{1}}\varphi_{0}ik \in \frac{1}{\lambda}\Sbf^2_\tau, \quad s\partial_{t}\varphi_{0} \in \frac{\ell'}{s^{1/2}\lambda}\Sbf^{3/2}_{\tau}
\]
\[
s\left(\partial^{3}_{x_{1}}\varphi_0\right)-2s^{2}\partial_{x_{1}}\varphi_{0}\partial_{x_{1}}^2\varphi_{0} -2s\partial_{x_{1}}^2\varphi_{0}ik -2s\partial_{x_{1}}\varphi_{0}k^2 \in \frac{1}{\lambda}\Sbf^3_\tau,
\quad s\partial_{t}\partial_{x_1}\varphi_{0}\in \frac{\ell'}{s^{1/2}\lambda}\Sbf^{3/2}_{\tau},
\]
\[
-s\nabla \varphi_0, -s\nabla \partial_{x_1} \varphi_0 \in \frac{1}{\lambda}\Sbf^1_\tau,
\]
so that, from \cref{Commutatorcorollary}, \cref{Continuitytheorem} and~\eqref{apriori3},
\begin{multline}\label{1348}
\left\| F^{(3)}\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +
\left\| F^{(4)}\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
\\
\lesssim \frac{1}{\lambda^2} \left(
\left\| \tau^2 \check{u} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\left\| \tau\partial_{x_1} \check{u} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\left\|\partial_{x_1}^2 \check{u} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
\right)
\\+\frac{1}{\lambda^2} \left(
\left\| \check{p} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\left\| \partial_{x_1} \check{p} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
\right)\lesssim \frac{1}{\lambda^2}\left(I_5(s,\lambda,\check u)+I_6(s,\lambda,\check p)\right).
\end{multline}
From~\eqref{ddeff45},
\begin{equation}\label{1332}
F^{(5)}=-\widetilde\chi^{\infty} \Op(\partial_t \chi)\partial_{x_1} \check{u}-\left(\widetilde\chi^{\infty}\right)' \Op(\partial_t \chi) \check{u}.
\end{equation}
From \cref{L02},
\[
\partial_{t} \chi \in \frac{\ell'}{(\lambda s)^{1/2} }\Sbf^{1/2}_{\tau}
\]
so that from \cref{Continuitytheorem} and~\eqref{1813},
\begin{multline}\label{1426}
\left\| F^{(5)} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
\lesssim
\lambda^{-1} \left(
\left\| \tau \partial_{x_1}\check{u} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\left\| \tau \check{u} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\left\| \partial_{x_1}^2\check{u} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
\right)
\\
\lesssim \lambda^{-2}I_5(s,\lambda,\check u).
\end{multline}
Finally, from~\eqref{ddeff6}, \eqref{apriori3} and~\eqref{lu11:34},
\[
\left\| F^{(6)} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
\lesssim \lambda^{-2} \widetilde{I}(\widetilde u, \widetilde p).
\]
Gathering~\eqref{ddeff}, \eqref{1206}, \eqref{1839}, \eqref{1348}, \eqref{1426} and the above estimate, we deduce the result.
\end{proof}


\begin{prop}\label{P02}
There exist $s_0>0$ and $\lambda_0>0$ such that the function $\widetilde f^{(2)}$ defined by~\eqref{defg} satisfies for $s\geq s_0(T^2+T^4)$ and for $\lambda\geq \lambda_0$,
\[
\left\| \partial_{x_1}\widetilde{f}^{(2)}\right\|_{L^2(0,T;H^1(\Omega^\infty))}^2 +\left\|\partial_{t}\widetilde{f}^{(2)}\right\|^{2}_{L^{2}(0,T;L^{2}(\Omega^{\infty}))}
\leq \lambda^{-1} \left(I_4(s,\lambda,\check \eta)+I_5(s,\lambda,\check u)+I_6(s,\lambda,\check p)+\widetilde{I}(\widetilde u, \widetilde p)
\right).
\]
\end{prop}
\begin{proof}
From~\eqref{defg}
\begin{equation}\label{1210}
\partial_{x_1} \widetilde f^{(2)}=G^{(1)}+G^{(2)}+G^{(3)}+G^{(4)},
\end{equation}
with
\begin{align}\label{ddegg1}
G^{(1)}&:=\partial_{x_1} \left(\widetilde\chi^{\infty} \Op(1-\chi)\left(e^{s\varphi_{0}}f^{(2)}\right)\right),
& G^{(2)}&:=\partial_{x_1} \left(s\partial_{x_1} \varphi_{0} \widetilde{u}^{\infty}_1\right),
\\\label{ddegg1bis}
G^{(3)}&:=- \partial_{x_1} [s\partial_{x_1} \varphi_{0},\widetilde\chi^{\infty} \Op(1-\chi)] \check u_1,
& G^{(4)}&:=\partial_{x_1}\left(\left(\widetilde\chi^{\infty}\right)' \widetilde{u}_1\right).
\end{align}
From~\eqref{ddegg1}, we have
\begin{align*}
G^{(1)}&=
\left(\widetilde\chi^{\infty}\right)' \Op(1-\chi)\left(e^{s\varphi_{0}}f^{(2)}\right) +\widetilde\chi^{\infty} \Op(1-\chi)\partial_{x_1}\left(e^{s\varphi_{0}}f^{(2)}\right),
\\
\partial_{x_1} G^{(1)}&=
\left(\widetilde\chi^{\infty}\right)'' \Op(1-\chi)\left(e^{s\varphi_{0}}f^{(2)}\right) +2\left(\widetilde\chi^{\infty}\right)' \Op(1-\chi)\partial_{x_1}\left(e^{s\varphi_{0}}f^{(2)}\right) +\widetilde\chi^{\infty} \Op(1-\chi)\partial_{x_1}^2\left(e^{s\varphi_{0}}f^{(2)}\right),
\\
\partial_{x_2} G^{(1)}&=
\left(\widetilde\chi^{\infty}\right)' \Op(1-\chi)\left(e^{s\varphi_{0}} \partial_{x_2}f^{(2)}\right) +\widetilde\chi^{\infty} \Op(1-\chi)\partial_{x_1}\left(e^{s\varphi_{0}} \partial_{x_2} f^{(2)}\right),
\end{align*}
with
\begin{align*}
\partial_{x_1}\left(e^{s\varphi_{0}}f^{(2)}\right)&=s\partial_{x_1}\varphi_0 e^{s\varphi_{0}} f^{(2)}+e^{s\varphi_{0}} \partial_{x_1}f^{(2)},
\\
\partial_{x_1}^2\left(e^{s\varphi_{0}}f^{(2)}\right)&= s\partial_{x_1}^2\varphi_0 e^{s\varphi_{0}} f^{(2)}+s^2\left(\partial_{x_1}\varphi_0\right)^2 e^{s\varphi_{0}} f^{(2)}+2s\partial_{x_1}\varphi_0 e^{s\varphi_{0}} \partial_{x_1}f^{(2)} +e^{s\varphi_{0}} \partial_{x_1}^2 f^{(2)},
\\
\partial_{x_1}\left(e^{s\varphi_{0}} \partial_{x_2} f^{(2)}\right)&= s\partial_{x_1}\varphi_0 e^{s\varphi_{0}} \partial_{x_2} f^{(2)}+e^{s\varphi_{0}} \partial_{x_1}\partial_{x_2} f^{(2)}.
\end{align*}
The above relations, combined with~\eqref{1519}, \eqref{1702}, with the properties of $\chi^\infty$ and with the periodicity of $u$ and $p$ in the $x_1$ variable, imply
\begin{multline*}
\left\| e^{s\varphi_{0}}f^{(2)} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\left\| \nabla \left(e^{s\varphi_{0}}f^{(2)}\right) \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 +\left\| \nabla \partial_{x_1}\left(e^{s\varphi_{0}}f^{(2)}\right) \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2
\\
\lesssim \iint_{(0,T)\times (0,2\pi) \times (0,1)} e^{2s\varphi_0} \left(
\left(\frac{\tau}{\lambda}\right)^4 \left|u\right|^2+ \left(\frac{\tau}{\lambda}\right)^2 \left|\nabla u\right|^2 + \left|\partial_{x_1}^2 u\right|^2+\left|\partial_{x_1}\partial_{x_2} u\right|^2
\right)\ \dx\,\dt
\\
\lesssim
\iint_{(0,T)\times \Omega^{\infty}} \left(
\left| \nabla^2 \check u \right|^2 + \left(\frac{\tau}{\lambda}\right)^2 \left| \nabla \check u \right|^2 + \left(\frac{\tau}{\lambda}\right)^4\left| \check u \right|^2
\right)\ \dx\,\dt.
\end{multline*}
Thus, using $1-\chi \in \Sbf^{0}_{\tau}$ (\cref{L01}) along with \cref{Continuitytheorem}, the property of $\chi^\infty$ and~\eqref{1813}, we deduce that
\begin{equation}\label{1540}
\left\| G^{(1)}\right\|_{L^2(0,T;H^1(\Omega^\infty))}^2
\leq \lambda^{-1} I_5(s,\lambda,\check u).
\end{equation}
Moreover, using~\eqref{1702}, we deduce
\begin{equation}\label{1105}
\left| G^{(2)} \right|+\left| \nabla G^{(2)} \right|
\lesssim \frac{\tau}{\lambda}
\left(\left| \widetilde{u}\right| + \left|\partial_{x_1} \widetilde{u}\right|+ \left|\partial_{x_1}^2 \widetilde{u}\right| + \left|\partial_{x_2} \widetilde{u}\right|+ \left|\partial_{x_1}\partial_{x_2} \widetilde{u}\right|\right).
\end{equation}
We deduce from the above relation and~\eqref{apriori3} that
\begin{equation}\label{1559}
\left\| G^{(2)}\right\|_{L^2(0,T;H^1(\Omega^\infty))}^2
\leq \lambda^{-2} \widetilde{I}(\widetilde u, \widetilde p).
\end{equation}
From~\eqref{ddegg1bis},
\[
\partial_{x_1} G^{(3)}=- \partial_{x_1}^2 [s\partial_{x_1} \varphi_{0},\widetilde\chi^{\infty} \Op(1-\chi)] \check u_1,
\quad
\partial_{x_2} G^{(3)}=- \partial_{x_1} [s\partial_{x_1} \varphi_{0},\widetilde\chi^{\infty} \Op(1-\chi)] \partial_{x_2} \check u_1.
\]

Thus from~\eqref{lu10:44}, \eqref{1813}, \cref{Commutatorcorollary} and \cref{Continuitytheorem},
\begin{equation}\label{18:17}
\left\| G^{(3)}\right\|_{L^2(0,T;H^1(\Omega^\infty))}^2
\lesssim \lambda^{-4} I_5(s,\lambda,\check u).
\end{equation}
From~\eqref{ddegg1bis},
\[
\partial_{x_1}G^{(4)}=\partial_{x_1}^2\left(\left(\widetilde\chi^{\infty}\right)' \widetilde{u}_1\right),
\quad
\partial_{x_2}G^{(4)}=\partial_{x_1}\left(\left(\widetilde\chi^{\infty}\right)' \partial_{x_2}\widetilde{u}_1\right),
\]
and thus, from~\eqref{apriori3} and~\eqref{lu11:34},
\[
\left\| G^{(4)} \right\|_{L^2(0,T;H^1(\Omega^\infty))}^2
\lesssim \lambda^{-2} \widetilde{I}(\widetilde u, \widetilde p).
\]
Gathering~\eqref{1210}, \eqref{1540}, \eqref{1559}, \eqref{18:17} and the above relation, we deduce the estimate for $\partial_{x_1}\widetilde{f}^{(2)}$. To estimate $\partial_{t}\widetilde{f}^{(2)}$, we derive~\eqref{defg} with respect to time:
\begin{equation}\label{18:32}
\partial_{t} \widetilde f^{(2)}=H^{(1)}+H^{(2)}+H^{(3)},
\end{equation}
with
\begin{align}\label{18:33}
H^{(1)}&:=- \widetilde\chi^{\infty} \Op(-\partial_{t} \chi)\left(e^{s\varphi_{0}}f^{(2)}\right) +\widetilde\chi^{\infty} \Op(1-\chi)\left(\left(s\partial_{t}\varphi_0 f^{(2)}+\partial_{t} f^{(2)}\right)e^{s\varphi_{0}}\right),
\\
H^{(2)}&:=\partial_{t} \left(s\partial_{x_1} \varphi_{0} \widetilde{u}^{\infty}_1+\left(\widetilde\chi^{\infty}\right)' \widetilde{u}_1\right),
\\\label{18:34}
H^{(3)}&:=- \partial_{t} [s\partial_{x_1} \varphi_{0},\widetilde\chi^{\infty} \Op(1-\chi)] \check u_1.
\end{align}
Combining~\eqref{1519}, \cref{L02}, \eqref{1702}, \cref{Continuitytheorem}, the property of $\chi^\infty$ and~\eqref{1813}, we deduce that
\begin{equation}\label{me09:18}
\left\| H^{(1)} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 \lesssim \lambda^{-1}I_{5}(s,\lambda,\check u).
\end{equation}
Using~\eqref{1519}, \eqref{1702} and~\eqref{apriori3}, we also find
\begin{equation}\label{me09:21}
\left\| H^{(2)}\right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 \leq \lambda^{-2} \widetilde{I}(\widetilde u, \widetilde p).
\end{equation}
For the last term, we write
\[
H^{(3)}=- [s\partial_{x_1}\partial_{t} \varphi_{0},\widetilde\chi^{\infty} \Op(1-\chi)] \check u_1 + [s\partial_{x_1} \varphi_{0},\widetilde\chi^{\infty} \Op(\partial_{t}\chi)] \check u_1 - [s\partial_{x_1} \varphi_{0},\widetilde\chi^{\infty} \Op(1-\chi)] \partial_{t} \check u_1.
\]
Combining~\eqref{lu10:44}, \cref{L02}, \cref{Commutatorcorollary} and \cref{Continuitytheorem}, we deduce
\[
\left\| H^{(3)} \right\|_{L^2(0,T;L^2(\Omega^\infty))}^2 \lesssim \lambda^{-1}I_{5}(s,\lambda,\check u).
\]
Gathering the above estimate, \eqref{me09:18} and~\eqref{me09:21} yields the result.
\end{proof}

\begin{prop}\label{P03}
There exist $s_0>0$ and $\lambda_0>0$ such that the function $\widetilde h$ defined by~\eqref{defh} satisfies for $s\geq s_0(T^2+T^4)$ and for $\lambda\geq \lambda_0$,
\[
\left\| \tau^{-1/2} \partial_t \partial_{x_1}\widetilde{h} \right\|_{L^2(0,T;L^2(\Gamma_1^\infty))}^2 +\left\| \partial_{x_1} \widetilde h \right\|_{L^2(0,T;H^{3/2}(\Gamma_1^\infty))}^2
\lesssim \lambda^{-1} I_4(s,\lambda,\check \eta).
\]
\end{prop}

\begin{proof}
From~\eqref{defh}
\begin{equation}\label{1902}
\partial_{x_1}\widetilde h =(\widetilde\chi^{\infty})' \Op(1-\chi)\widetilde h^{(1)} +\widetilde\chi^{\infty}\Op(1-\chi)\widetilde h^{(2)},
\end{equation}
with
\[
\widetilde h^{(1)}:=\partial_{t}\check \eta - s\partial_{t}\varphi_{0}\check \eta,\quad
\widetilde h^{(2)}:=\partial_{x_1}\widetilde h^{(1)} =\partial_{t}\partial_{x_1}\check\eta-s\left(\partial_t \partial_{x_1}\varphi_0\right)\check \eta -s\left(\partial_t \varphi_0\right)\partial_{x_1}\check \eta.
\]
Applying \cref{L03} and using~\eqref{1902}, we deduce that
\begin{multline*}
\left\| \partial_{x_1} \widetilde h \right\|_{L^2(0,T;H^{3/2}(\Gamma_1^\infty))}^2
\lesssim
\left\| \Op(1-\chi)\widetilde h^{(1)} \right\|_{L^2(0,T;H^{3/2}(\Gamma_1^\infty))}^2 +\left\| \Op(1-\chi)\widetilde h^{(2)} \right\|_{L^2(0,T;H^{3/2}(\Gamma_1^\infty))}^2
\\
\lesssim
\left\| \tau^{-1/2} \Op(1-\chi) \partial_{x_1}^2 \widetilde h^{(1)} \right\|_{L^2(0,T;L^{2}(\Gamma_1^\infty))}^2 +\left\| \tau^{-1/2} \Op(1-\chi) \partial_{x_1}^2 \widetilde h^{(2)} \right\|_{L^2(0,T;L^{2}(\Gamma_1^\infty))}^2.
\end{multline*}
Then, using that $1-\chi \in \Sbf^{0}_{\tau}$, \eqref{1702} and \cref{Continuitytheorem}, we find
\begin{multline*}
\left\| \partial_{x_1} \widetilde h \right\|_{L^2(0,T;H^{3/2}(\Gamma_1^\infty))}^2
\lesssim
\left\| \tau^{-1/2} \partial_{t}\partial_{x_1}^3\check \eta \right\|_{L^2(0,T;L^{2}(\Gamma_1^\infty))}^2
\\
+\left\| \tau^{-1/2} \partial_{t}\partial_{x_1}^2\check \eta \right\|_{L^2(0,T;L^{2}(\Gamma_1^\infty))}^2 +\sum_{j=0}^3 \left\| \frac{\tau^{3/2}}{\lambda^{3/2}}\partial_{x_1}^j\check \eta \right\|_{L^2(0,T;L^{2}(\Gamma_1^\infty))}^2.
\end{multline*}

From~\eqref{1812}, we deduce from the above relation that
\begin{equation}\label{1904}
\left\| \partial_{x_1} \widetilde h \right\|_{L^2(0,T;H^{3/2}(\Gamma_1^\infty))}^2 \lesssim \lambda^{-1}I_4(s,\lambda,\check \eta).
\end{equation}
By differentiating~\eqref{1902} with respect to $t$, we obtain
\begin{equation}\label{1903}
\partial_t \partial_{x_1}\widetilde h =-(\widetilde\chi^{\infty})' \Op(\partial_t \chi)\widetilde h^{(1)} +(\widetilde\chi^{\infty})' \Op(1-\chi)\partial_t \widetilde h^{(1)} -\widetilde\chi^{\infty}\Op(\partial_t \chi)\widetilde h^{(2)} +\widetilde\chi^{\infty}\Op(1-\chi)\partial_t\widetilde h^{(2)}.
\end{equation}
Applying \cref{L02}, \cref{Continuitytheorem} and~\eqref{1702}, we have for $s\geq s_0(T^2+T^4)$ and $\lambda\geq \lambda_0$,
\begin{multline}\label{1911}
\left\| \tau^{-1/2} \partial_t \partial_{x_1}\widetilde{h} \right\|_{L^2(0,T;L^2(\Gamma_1^\infty))}^2
\lesssim
\left\| \frac{\tau^{3/2}}{\lambda^{3/2}}\partial_{x_1}\check\eta \right\|_{L^2(0,T;L^2(\Gamma_1^\infty))}^2 +\left\| \frac{\tau^{3/2}}{\lambda^{3/2}}\check\eta \right\|_{L^2(0,T;L^2(\Gamma_1^\infty))}^2
\\
\begin{aligned}[b]
&+\left\| \tau^{-1/2}\partial_{t}^2\check \eta \right\|_{L^2(0,T;L^2(\Gamma_1^\infty))}^2 +\left\| \tau^{-1/2}\partial_{t}^2\partial_{x_1}\check \eta \right\|_{L^2(0,T;L^2(\Gamma_1^\infty))}^2
\\
&+\left\| \frac{\tau}{\lambda^{3/2}}\partial_{x_1}\partial_t \check\eta \right\|_{L^2(0,T;L^2(\Gamma_1^\infty))}^2 +\left\| \frac{\tau}{\lambda^{3/2}}\partial_{t} \check\eta \right\|_{L^2(0,T;L^2(\Gamma_1^\infty))}^2.
\end{aligned}
\end{multline}
From~\eqref{1812}, we deduce from the above relation that
\begin{equation}\label{0953}
\left\| \tau^{-1/2} \partial_t \partial_{x_1}\widetilde{h} \right\|_{L^2(0,T;L^2(\Gamma_1^\infty))}^2
\lesssim \lambda^{-1}I_4(s,\lambda,\check \eta).
\end{equation}
\end{proof}

The proof of \cref{P05} consists now in combining~\eqref{1619} with \cref{P01}, \cref{P02} and \cref{P03}. In the next section, we show the observability result from the above relation.

\section{Proof of the observability}\label{sec_obs}
This section is devoted to the proof of \cref{T01}. We first remove in~\eqref{1014} the local terms in $p$ and in $\partial_{t}^2\eta$. Setting
\begin{align}\label{1131123qu}
\varphi_1(t) &:= \frac{1}{\ell(t)^2}(e^{8\lambda\Psi}-e^{10\lambda\Psi}),&
\xi_1(t)&:=\frac{1}{\ell(t)^2}e^{8\lambda\Psi},
\\ \label{1131123ter}
\varphi_2(t) &:= \frac{1}{\ell(t)^2}(e^{(9\lambda+\mu_0)\Psi}-e^{10\lambda\Psi}),&
\xi_2(t)&:=\frac{1}{\ell(t)^2}e^{(9\lambda+\mu_0)\Psi},
\end{align}
we have (see~\eqref{1131123})
\[
\varphi_1(t)\leq \varphi(t,\cdot\,)\leq \varphi_2(t), \quad \xi_1(t)\leq \xi(t,\cdot\,)\leq \xi_2(t) \quad (t\in (0,T)).
\]
Let us set
\begin{equation}\label{1745}
\rho_0:=\lambda \tau^2 e^{s\varphi_1}=\lambda^3 e^{16\lambda \Psi} \frac{s^2}{\ell^4} e^{s\varphi_1},
\end{equation}
\begin{equation}\label{ma16:31-1}
\rho_{1}:=s^{11/2}\lambda^{-7} \xi_{2}^{11/2} e^{4s\varphi_{2}-3s\varphi_1}, \quad
\rho_{2}:=s^{9}\lambda^{-1} \xi_2^{9} e^{4s\varphi_2-3s\varphi_1}.
\end{equation}
Then we have the following result.

\begin{prop}\label{P06}
There exist $s_0>0$ and $\lambda_0>0$ such that for any $s\geq s_0(T^2+T^4)$ and for any $\lambda\geq \lambda_0$, any smooth solution of~\eqref{ns0.2} satisfies
\begin{equation}\label{1201}
\int_0^T \rho_0^2 \left\| [u, \eta, \partial_t \eta]\right\|_{\mathcal{H}}^2\ \dt
\lesssim
\iint_{(0,T)\times\mathcal J} \rho_{1}^2 |\partial_t \eta|^{2} \ \dt\, \dx_1
\\
+\iint_{(0,T)\times \omega} \rho_{2}^2 |u|^{2}\ \dt\, \dx.
\end{equation}
\end{prop}
\begin{proof}
Using~\eqref{ma10:17} and applying the Poincar\'e--Wirtinger inequality, we deduce that
\begin{equation*}
\iint_{(0,T)\times \omega_1} s^{3}\lambda^{4}\xi^{3} e^{2s\varphi} |p|^{2}\ \dt\, \dx
\lesssim
\iint_{(0,T)\times \omega_1} s^{3}\lambda^{4}\xi_2^{3} e^{2s\varphi_2} |\nabla p|^{2}\ \dt\, \dx
\end{equation*}
and with~\eqref{ns0.3},
\begin{equation}\label{1037}
\iint_{(0,T)\times \omega_1} s^{3}\lambda^{4}\xi^{3} e^{2s\varphi} |p|^{2}\ \dt\, \dx
\lesssim
\iint_{(0,T)\times \omega_1} s^{3}\lambda^{4}\xi_2^{3} e^{2s\varphi_2} \left(|\partial_t u|^{2}+|\Delta u|^{2}\right)\ \dt\, \dx.
\end{equation}
From~\eqref{1745}, \eqref{1812} and~\eqref{1813}, we have
\begin{equation}\label{1748}
\int_0^T \rho_0(t)^2 \left(\left\| u(t)\right\|_{L^2(\Omega)}^2+\left\| \eta(t)\right\|_{H^2(\mathcal{I})}^2+\left\| \partial_t \eta(t)\right\|_{L^2(\mathcal{I})}^2 \right) \ \dt
\lesssim I_4(s,\lambda,\check \eta)+I_5(s,\lambda,\check u).
\end{equation}
Combining~\eqref{1014}, \eqref{1748} and~\eqref{1037}, we deduce
\begin{multline}\label{1132}
\left\| \rho_0 u\right\|_{L^2(0,T;L^2(\Omega))}^2 +\left\| \rho_0 \partial_t \eta\right\|_{L^2(0,T;L^2(\Gamma_1))}^2 +\left\| \rho_0 \eta\right\|_{L^2(0,T;H^2(\Gamma_1))}^2
\\
\begin{aligned}[b]
&\lesssim
\lambda \iint_{(0,T)\times\mathcal J_1} e^{2s\varphi_{0}} \left(s^{10}\xi_{0}^{10}|\eta|^{2}+s^{2}\xi_{0}^{2}|\partial_{t}^2\eta|^{2}
\right)\ \dt\, \dx_1
\\
&\qquad+ \iint_{(0,T)\times\omega_1} s^{4}\lambda^{6}\xi^{4}e^{2s\varphi} |u|^{2} \ \dt\,\dx +\iint_{(0,T)\times \omega_1} s^{3}\lambda^{4}\xi_2^{3} e^{2s\varphi_2} \left(|\partial_t u|^{2}+|\Delta u|^{2}\right)\ \dt\, \dx.
\end{aligned}
\end{multline}
Now we set
\begin{equation}\label{0955}
\rho_{3}(t):=\lambda^3 e^{6\lambda \Psi} \frac{s^{1/2}}{\ell(t)} e^{s\varphi_1(t)},
\quad
\rho_{4}(t):=\lambda^3 e^{-4\lambda \Psi} s^{-1} \ell(t)^2 e^{s\varphi_1(t)}.
\end{equation}
We have $\rho_{3}, \rho_{4}\in C^1([0,T])$, $\rho_{3}(0)=\rho_{4}(0)=0$ and
\begin{equation}\label{1642}
\left| \rho_{3}'\right| \lesssim \rho_0
\quad \text{and} \quad
\left|\rho_{4}'\right| \lesssim \rho_{3}.
\end{equation}
We recall that~\eqref{ns0.3} can be written as~\eqref{0907}. Then, we deduce
\begin{equation}\label{0909}
\frac{\dd}{\dt} \rho_{3}
\begin{bmatrix}
u\\ \eta \\ \partial_t \eta
\end{bmatrix} =\mathcal{A}\rho_{3}
\begin{bmatrix}
u\\ \eta \\ \partial_t \eta
\end{bmatrix} +\rho_{3}'
\begin{bmatrix}
u\\ \eta \\ \partial_t \eta
\end{bmatrix}
\quad \text{in} \ (0,T), \quad
\begin{bmatrix}
\rho_{3} u\\ \rho_{3} \eta \\ \rho_{3} \partial_t \eta
\end{bmatrix}(0)=0.
\end{equation}
From~\eqref{1642}, \eqref{1748} and~\eqref{ma11:10},
\begin{multline}\label{1630}
\left\| \rho_{3} \partial_t u\right\|_{L^2(0,T;L^2(\Omega))} +\left\| \rho_{3} u\right\|_{L^2(0,T;H^2(\Omega))}\\
+\left\| \rho_{3} \partial_t^2 \eta\right\|_{L^2(0,T;L^2(\Gamma_1))}
+\left\| \rho_{3} \partial_t \eta\right\|_{L^2(0,T;H^2(\Gamma_1))} +\left\| \rho_{3} \eta\right\|_{L^2(0,T;H^4(\Gamma_1))}
\\
\lesssim
\left\| \rho_0 u\right\|_{L^2(0,T;L^2(\Omega))}^2 +\left\| \rho_0 \partial_t \eta\right\|_{L^2(0,T;L^2(\Gamma_1))}^2 +\left\| \rho_0 \eta\right\|_{L^2(0,T;H^2(\Gamma_1))}^2.
\end{multline}
Then, we deduce from~\eqref{0907} that
\begin{equation}\label{ma11:28}
\frac{\dd}{\dt} \left(\rho_{4}
\frac{\dd}{\dt}
\begin{bmatrix}
u\\ \eta \\ \partial_t \eta
\end{bmatrix}\right) =\mathcal{A}\left(\rho_{4}
\frac{\dd}{\dt}
\begin{bmatrix}
u\\ \eta \\ \partial_t \eta
\end{bmatrix}\right) +\rho_{4}'
\frac{\dd}{\dt}
\begin{bmatrix}
u\\ \eta \\ \partial_t \eta
\end{bmatrix}
\quad \text{in} \ (0,T), \quad
\left(\rho_{4}
\frac{\dd}{\dt}
\begin{bmatrix}
u\\ \eta \\ \partial_t \eta
\end{bmatrix}\right)(0)=0.
\end{equation}
From~\eqref{1642}, \eqref{1748}, \eqref{1630} and~\eqref{ma11:10},
\begin{multline}\label{1657-1}
\left\| \rho_{4} \partial_t^2 u\right\|_{L^2(0,T;L^2(\Omega))} +\left\| \rho_{4} \partial_t u\right\|_{L^2(0,T;H^2(\Omega))}
\\
+\left\| \rho_{4} \partial_t^3 \eta\right\|_{L^2(0,T;L^2(\Gamma_1))} +\left\| \rho_{4} \partial_t^2 \eta\right\|_{L^2(0,T;H^2(\Gamma_1))} +\left\| \rho_{4} \partial_t \eta\right\|_{L^2(0,T;H^4(\Gamma_1))}
\\
\lesssim
\left\| \rho_0 u\right\|_{L^2(0,T;L^2(\Omega))}^2 +\left\| \rho_0 \partial_t \eta\right\|_{L^2(0,T;L^2(\Gamma_1))}^2 +\left\| \rho_0 \eta\right\|_{L^2(0,T;H^2(\Gamma_1))}^2.
\end{multline}
Then, from the standard elliptic regularities for the stationary Stokes system (\cite[Proposition~2.2 p.~33]{Temam}) and for $A_1$, we have moreover
\begin{multline}\label{1657}
\left\| \rho_{4} u\right\|_{L^2(0,T;H^4(\Omega))} +\left\| \rho_{4} \eta\right\|_{L^2(0,T;H^6(\Gamma_1))}
\\
\lesssim
\left\| \rho_0 u\right\|_{L^2(0,T;L^2(\Omega))}^2 +\left\| \rho_0 \partial_t \eta\right\|_{L^2(0,T;L^2(\Gamma_1))}^2 +\left\| \rho_0 \eta\right\|_{L^2(0,T;H^2(\Gamma_1))}^2.
\end{multline}
By integration by parts, we obtain
\begin{multline*}
\iint_{(0,T)\times\mathcal J_1} s^{2}\lambda \xi_{0}^{2} e^{2s\varphi_{0}}|\partial_{t}^2\eta|^{2} \ \dt\, \dx_1 =\frac{1}{2}\iint_{(0,T)\times\mathcal J_1} \partial_t^2\left(s^{2}\lambda \xi_{0}^{2} e^{2s\varphi_{0}}\right) |\partial_{t}\eta|^{2} \ \dt\, \dx_1
\\
-\iint_{(0,T)\times\mathcal J_1} s^{2}\lambda \xi_{0}^{2} e^{2s\varphi_{0}}\partial_{t}^3\eta \partial_t \eta \ \dt\, \dx_1.
\end{multline*}
Using~\eqref{1700}, we deduce that for $s\geq s_0 (T^2+T^4)$, for any $\varepsilon>0$, there exists $C>0$ such that
\begin{multline}\label{1055}
\iint_{(0,T)\times\mathcal J_1} s^{2}\lambda \xi_{0}^{2} e^{2s\varphi_{0}}|\partial_{t}^2\eta|^{2} \ \dt\, \dx_1
\leq C\iint_{(0,T)\times\mathcal J_1} s^{5}\lambda \xi_{0}^{5} e^{2s\varphi_{0}} |\partial_{t}\eta|^{2} \ \dt\, \dx_1
\\
+\varepsilon \left\| \rho_{4} \partial_t^3 \eta\right\|_{L^2(0,T;L^2(\Gamma_1))}^2 +C \iint_{(0,T)\times\mathcal J_1} s^{6}\lambda^{-4} \xi_{0}^{6} e^{4s\varphi_{0}-2s\varphi_1} |\partial_{t}\eta|^{2} \ \dt\, \dx_1
\\
\leq
\varepsilon \left\| \rho_{4} \partial_t^3 \eta\right\|_{L^2(0,T;L^2(\Gamma_1))}^2 +C \iint_{(0,T)\times\mathcal J_1} s^{6}\lambda^{-4} \xi_{0}^{6} e^{4s\varphi_{0}-2s\varphi_1} |\partial_{t}\eta|^{2} \ \dt\, \dx_1.
\end{multline}
Then, we integrate by parts the last term and we obtain that for any $\varepsilon>0$, there exists $C>0$ such that
\begin{multline}\label{1106}
\iint_{(0,T)\times\mathcal J_1} s^{6}\lambda^{-4} \xi_{0}^{6} e^{4s\varphi_{0}-2s\varphi_1} |\partial_{t}\eta|^{2} \ \dt\, \dx_1
\\
\leq \varepsilon \left\| \rho_{3} \partial_t^2 \eta\right\|_{L^2(0,T;L^2(\Gamma_1))}^2 +C\iint_{(0,T)\times\mathcal J_1} s^{11}\lambda^{-14} \xi_{0}^{11} e^{8s\varphi_{0}-6s\varphi_1} |\eta|^{2} \ \dt\, \dx_1.
\end{multline}
Similarly, for any $\varepsilon>0$, there exists $C>0$ such that
\begin{multline}\label{1139}
\iint_{(0,T)\times \omega_1} s^{3}\lambda^{4}\xi_2^{3} e^{2s\varphi_2} |\partial_t u|^{2}\ \dt\, \dx
\leq C\iint_{(0,T)\times \omega_1} s^{6}\lambda^{4}\xi_2^{6} e^{2s\varphi_2} |u|^{2}\ \dt\, \dx
\\
+\varepsilon \left\| \rho_{4} \partial_t^2 u\right\|_{L^2(0,T;L^2(\Omega))}^2 + C\iint_{(0,T)\times \omega_1} s^{8}\lambda^{2}\xi_2^{8} e^{4s\varphi_2-2s\varphi_1} |u|^{2}\ \dt\, \dx
\\
\leq \varepsilon \left\| \rho_{4} \partial_t^2 u\right\|_{L^2(0,T;L^2(\Omega))}^2 + C\iint_{(0,T)\times \omega_1} s^{8}\lambda^{2}\xi_2^{8} e^{4s\varphi_2-2s\varphi_1} |u|^{2}\ \dt\, \dx.
\end{multline}
Finally, we consider a nonnegative smooth function $\chi_1$ with compact support in $\omega$ and such that $\chi_1\equiv 1$ in $\omega_1$. Then by integrating by parts,
\begin{multline}\label{1239}
\iint_{(0,T)\times \omega_1} s^{3}\lambda^{4}\xi_2^{3} e^{2s\varphi_2} |\Delta u|^{2} \ \dt\, \dx
\\
\leq
\iint_{(0,T)\times \omega} \chi_1 s^{3}\lambda^{4}\xi_2^{3} e^{2s\varphi_2} |\Delta u|^{2} \ \dt\, \dx =
\iint_{(0,T)\times \omega} s^{3}\lambda^{4}\xi_2^{3} e^{2s\varphi_2} \Delta\left(\chi_1 \Delta u\right) u \ \dt\, \dx
\\
\leq \varepsilon \left\| \rho_{4} u\right\|_{L^2(0,T;H^4(\Omega))}^2 + C\iint_{(0,T)\times \omega} s^{8}\lambda^{2}\xi_2^{8} e^{4s\varphi_2-2s\varphi_1} |u|^{2}\ \dt\, \dx.
\end{multline}

Gathering~\eqref{1132}, \eqref{1055}, \eqref{1106}, \eqref{1139}, and~\eqref{1239}, and using~\eqref{1630}, \eqref{1657-1} and~\eqref{1657} we deduce
\begin{multline}\label{1133}
\int_0^T \rho_0(t)^2 \left(\left\| u(t)\right\|_{L^2(\Omega)}^2+\left\| \eta(t)\right\|_{H^2(\mathcal{I})}^2+\left\| \partial_t \eta(t)\right\|_{L^2(\mathcal{I})}^2 \right) \ \dt
\\
\lesssim
\iint_{(0,T)\times\mathcal J_1} \rho_{1}^2 |\eta|^{2} \ \dt\, \dx_1 + \iint_{(0,T)\times \omega} \rho_5^2 |u|^{2}\ \dt\, \dx,
\end{multline}
with $\rho_{1}$ defined by~\eqref{ma16:31-1} and with
\begin{equation}\label{ma16:31}
\rho_5:=s^{4}\lambda \xi_2^{4} e^{2s\varphi_2-s\varphi_1}.
\end{equation}

To end the proof of \cref{P06}, we need to replace in the above estimate the observation by $\eta$ with an observation by $\partial_{t}\eta$. This is done by using the smoothing effet of the parabolic system~\eqref{0907}. More precisely, we apply~\eqref{1133} to $(\partial_t u, \partial_t \eta, \partial_t^2 \eta)$ and we deduce
\begin{multline}\label{1140}
\int_0^T \rho_0(t)^2 \left(\left\| \partial_t u(t)\right\|_{L^2(\Omega)}^2+\left\| \partial_t \eta(t)\right\|_{H^2(\mathcal{I})}^2+\left\| \partial_t^2 \eta(t)\right\|_{L^2(\mathcal{I})}^2 \right) \ \dt
\\
\lesssim
\iint_{(0,T)\times\mathcal J} \rho_{1}^2 |\partial_t \eta|^{2} \ \dt\, \dx_1 + \iint_{(0,T)\times \omega} \rho_5^2 |\partial_t u|^{2}\ \dt\, \dx.
\end{multline}
On the other hand, using~\eqref{0907} and the fact that $0\in \rho(\mathcal{A})$ (see, for instance, \cite[Proposition~3.5]{MR3619065}),
\[
\left\| \partial_t [u, \eta, \partial_t \eta]\right\|_{\mathcal{H}} =\left\| \mathcal{A}[u, \eta, \partial_t \eta]\right\|_{\mathcal{H}} \geq c \left\| [u, \eta, \partial_t \eta]\right\|_{\mathcal{H}}.
\]
Combining the above estimate and~\eqref{1140} implies
\begin{equation}\label{1147}
\int_0^T \rho_0^2 \left\| [u, \eta, \partial_t \eta]\right\|_{\mathcal{H}}^2\ \dt
\lesssim
\iint_{(0,T)\times\mathcal J} \rho_{1}^2 |\partial_t \eta|^{2} \ \dt\, \dx_1 + \iint_{(0,T)\times \omega} \rho_5^2 |\partial_t u|^{2}\ \dt\, \dx.
\end{equation}
We integrate by parts the last term: recalling~\eqref{ma16:31}, we obtain that for any $\varepsilon>0$, there exists $C>0$ such that
\begin{multline}\label{1156}
\iint_{(0,T)\times \omega_1} \rho_5^2 |\partial_t u|^{2}\ \dt\, \dx
\leq C\iint_{(0,T)\times \omega_1} s^{11}\lambda^{2}\xi_2^{11} e^{4s\varphi_2-2s\varphi_1} |u|^{2}\ \dt\, \dx
+\varepsilon \left\| \rho_{4} \partial_t^2 u\right\|_{L^2(0,T;L^2(\Omega))}^2\\
+C\iint_{(0,T)\times \omega_1} s^{18}\lambda^{-2} \xi_2^{18} e^{8s\varphi_2-6s\varphi_1} |u|^{2}\ \dt\, \dx.
\end{multline}
We deduce~\eqref{1201} by combining~\eqref{1147}, \eqref{1156} and~\eqref{1657}.
\end{proof}

Using \cref{P06}, one can deduce \cref{T01}:
\begin{proof}[Proof of \cref{T01}]
We fix $\lambda=\lambda_0$ and $s=s_0(T^2+T^4)$ in~\eqref{1745} and~\eqref{ma16:31-1}. In particular the constants that follows may depend on $\lambda_0$ and $s_0$. Then we deduce from~\eqref{1201} that
\begin{equation}\label{ma22:40}
\int_{T/4}^{3T/4} \rho_0^2 \left\| [u, \eta, \partial_t \eta]\right\|_{\mathcal{H}}^2\ \dt
\lesssim
\iint_{(0,T)\times\mathcal J} \rho_{1}(t)^2 |\partial_t \eta|^{2} \ \dt\, \dx_1
\\
+\iint_{(0,T)\times \omega} \rho_{2}(t)^2 |u|^{2}\ \dt\, \dx.
\end{equation}
From~\eqref{1745} and~\eqref{ma16:31-1}, there exists $C>0$ such that
\[
\frac{C}{T}e^{-C/T^2}\leq \rho_0 \quad \text{in} \ \left(\frac{T}{4},\frac{3T}{4}\right),
\]
and
\[
\rho_{1}\lesssim 1, \quad \rho_{2}\lesssim 1.
\]
Since $\mathcal{A}$ is the generator of a semigroup of contractions (see, for instance, \cite[Proposition~3.4]{MR3619065}), we deduce the result from the above relations.
\end{proof}


\section{Proof of \cref{T02}}\label{sec_proof}

We give here a sketch of the proof of \cref{T02}. First we construct a change of variables to write~\eqref{tak2.3} in a cylindrical domain, then we use the ``source term method'' and \cref{T01} to show \cref{T02} by a fixed-point argument.

\subsection{Change of variables}\label{sec_chg}

We can assume that for some $\delta >0$
\[
\omega \subset \mathcal{I}\times (0,1-\delta).
\]
Let us consider a smooth function $\theta\in C^\infty([0,1];[0,1])$ with compact support in $(1-\delta,1]$ and such that $\theta\equiv 1$ in $[1-\delta/2,1]$. We consider the change of variables
\begin{equation}\label{defX}
X(t,\cdot\,) : \Omega \to \Omega_{\zeta(t)}, \quad (y_1,y_2) \mapsto (y_1,y_2+\theta(y_2) \zeta (t,y_1))
\end{equation}
that is a diffeomorphism if
\begin{equation}\label{me12:28}
\| \theta'\|_{L^\infty(0,1)} \left\| \zeta \right\|_{L^\infty(0,T;L^\infty(\mathcal{I}))}\leq \frac 12.
\end{equation}
We denote by $Y(t,\cdot\,)$ the inverse of $X(t,\cdot\,)$.

We write
\[
W(t,y):=\Cof(\nabla X)^*(t,y) w(t,X(t,y)), \quad \Pi(t,y):=\pi(t,X(t,y))
\]
\[
X^0:=X(0,\cdot\,), \quad W^{0}:=\Cof(\nabla X^0)^* w^0\circ X^0.
\]
We also write
\begin{equation}\label{defa}
a:=\Cof(\nabla Y)^*.
\end{equation}
After some standard calculations (see, for instance, \cite{plat}) \eqref{tak2.3} is transformed into
\begin{equation}\label{me12:00}
\begin{cases}
\partial_t W-\div \mathbb{T}(W,\Pi) = 1_{\omega} f+\mathbb{F}_\zeta(W,\Pi) & t>0, \ x\in \Omega,\\
\div W = 0 & t>0, \ x\in \Omega,\\
W= \partial_t \zeta e_2 & t>0, \ x\in \Gamma_1,\\
W = 0 & t>0, \ x\in \Gamma_{0},\\
\partial_{tt} \zeta + A_1 \zeta+A_2 \partial_{t} \zeta =P_{L^2_0(\mathcal{I})}\left(\Pi +1_{\mathcal{J}} g+\mathbb{G}_\zeta(W)\right)& t>0, \ x_1\in \mathcal{I},
\end{cases}
\end{equation}
\begin{equation}\label{me12:01}
W(0,\cdot\,)=W^{0} \quad \text{in} \ \Omega,\quad \zeta(0,\cdot\,)=\zeta_1^0, \quad \partial_t\zeta (0,\cdot\,)=\zeta_2^0 \quad \text{in} \ \mathcal{I},
\end{equation}
with
\begin{multline}\label{me22:00}
\left[\mathbb{F}_\zeta(W,\Pi)\right]_i:= -\sum_{k} \partial_t a_{i,k}(X) W_k +\left(a_{i,k}(X)-\delta_{i,k}\right) \partial_t W_k-\sum_{k,l} a_{i,k}(X) \frac{\partial W_k}{\partial y_l} \frac{\partial Y_l}{\partial t}(X)
\\
\begin{aligned}[b]
&+\sum_{k,j} \frac{\partial^2 a_{i,k}}{\partial x_j^2}(X) W_k +2\sum_{k,j,l}\frac{\partial a_{i,k}}{\partial x_j}(X) \frac{\partial W_k}{\partial y_l}\frac{\partial Y_l}{\partial x_j}(X) +\sum_{k,j,l} a_{i,k}(X) \frac{\partial W_k}{\partial y_l}\frac{\partial^2 Y_l}{\partial x_j^2}(X)
\\
&+\sum_{k,j,l,m} \left(a_{i,k}(X)\frac{\partial Y_l}{\partial x_j}(X)\frac{\partial Y_m}{\partial x_j}(X)-\delta_{i,k}\delta_{l,j}\delta_{m,j}\right)
\frac{\partial W_k}{\partial y_l\partial y_m} -\sum_{l} \left(\frac{\partial Y_l}{\partial x_i}(X)-\delta_{l,i}\right) \frac{\partial \Pi}{\partial y_l}
\end{aligned}
\\
-\sum_{j,m,k} a_{j,m}(X) \frac{\partial a_{i,k}}{\partial x_j}(X) W_m W_k -\frac{1}{\det (\nabla X)}\sum_{m,k} a_{i,k}(X) W_m \frac{\partial W_k}{\partial y_m},
\end{multline}
\begin{equation}\label{me22:01}
\mathbb{G}_\zeta(W):=-\left(\partial_{x_1}\zeta+\left(\partial_{x_1}\zeta\right)^2\right) \left.\frac{\partial W_1}{\partial x_2}\right|_{x_2=1}.
\end{equation}
Then, we can write \cref{T02} as follows:

\begin{theo}\label{T03}
Assume $T>0$, $\omega\Subset \Omega$ and $\mathcal{J}\Subset \mathcal{I}$ are nonempty open sets. There exists $R_0>0$ such that for any $[W^0, \zeta_1^0, \zeta_2^0]\in \mathcal{V}$ with
\begin{equation}\label{je15:03}
\left\| [W^0, \zeta_1^0, \zeta_2^0]\right\|_{\mathcal{V}} \leq R_0,
\end{equation}
there exists a control
\[
(f,g)\in L^2(0,T;L^2(\omega))\times L^2(0,T;L^2(\mathcal{J}))
\]
such that the solution of~\eqref{me12:00}, \eqref{me12:01}, \eqref{me22:00} and~\eqref{me22:01} satisfies
\[
\zeta(T,\cdot\,)=0, \quad \partial_t \zeta(T,\cdot\,)=0 \quad \text{in} \ \mathcal{I},\quad W(T,\cdot\,)=0 \quad \text{in} \ \Omega.
\]
\end{theo}

\subsection{The fixed point argument} Using the notation of \cref{sec_func}, the result of \cref{T01} states the existence of $k_T$ satisfying~\eqref{je11:14} such that for any $
\begin{bmatrix} u^0, \eta_1^0, \eta_2^0
\end{bmatrix}$,
\begin{equation}\label{je11:10}
\left\| e^{T\mathcal{A}^*}
\begin{bmatrix} u^0\\ \eta_1^0 \\ \eta_2^0
\end{bmatrix} \right\|_{\mathcal{H}}^2 \leq k_T^2 \int_0^T \left\| \mathcal{B}^* e^{t\mathcal{A}^*}
\begin{bmatrix} u^0\\ \eta_1^0 \\ \eta_2^0
\end{bmatrix}\right\|_{L^2(\omega)\times L^2(\mathcal{J})}^2 \ \dt.
\end{equation}
From standard results (see, for instance, \cite[Theorem~11.2.1, p.~357]{TucsnakWeiss}), this yields the null-controllability of~\eqref{ns0.1}. Using the ``source term method'' (see, \cite{Yuning}), one can improve this result. Let us consider the following weight functions
\begin{equation}\label{je11:59}
\sigma_1(t):=e^{-\frac{C_1}{(T-t)^2}}, \quad
\sigma_2(t):=e^{-\frac{C_2}{(T-t)^2}}, \quad
\sigma_3(t):=e^{-\frac{C_3}{(T-t)^2}}
\end{equation}
and the corresponding spaces (for $\sigma=\sigma_1,\sigma_2$ or $\sigma_3$)
\[
L^p_{\sigma}(0,T;\mathcal{X}):=\left\{ f/\sigma \in L^p(0,T;\mathcal{X})\right\},
\]
\[
C^{\alpha}_\sigma([0,T];\mathcal{X}):=\left\{ f/\sigma \in C^\alpha([0,T];\mathcal{X})\right\},
\]
\[
H^s_\sigma(0,T;\mathcal{X}):=\left\{ f/\sigma \in H^s(0,T;\mathcal{X})\right\},
\]
for $p\geq 1$, $k\in \mathbb{N}$, $s\in \mathbb{R}_+$ and $\mathcal{X}$ a Banach space. The abstract result proved in~\cite{Yuning} yields the following result:

\begin{prop}\label{P07}
Assume~\eqref{je11:10} with~\eqref{je11:14}. Then there exist $\sigma_1, \sigma_2, \sigma_3$ as in~\eqref{je11:59} and a bounded~map
\[
\mathbb{E}_T : \mathcal{V}\times L^2_{\sigma_1}(0,T;L^2(\Omega)\times L^2(\mathcal{I})) \to L^2_{\sigma_2}(0,T;L^2(\omega)\times L^2(\mathcal{J}))
\]
such that for any $[W^0, \zeta_1^0, \zeta_2^0]\in \mathcal{V}$ and for any $(F,G)\in L^2_{\sigma_{1}}(0,T;L^2(\Omega)\times L^2(\mathcal{I}))$, the solution of
\begin{equation}\label{je15:00}
\begin{cases}
\partial_t W-\div \mathbb{T}(W,\Pi) = 1_{\omega} f+F & t>0, \ x\in \Omega,\\
\div W = 0 & t>0, \ x\in \Omega,\\
W= \partial_t \zeta e_2 & t>0, \ x\in \Gamma_1,\\
W = 0 & t>0, \ x\in \Gamma_{0},\\
\partial_{tt} \zeta + A_1 \zeta+A_2 \partial_{t} \zeta =P_{L^2_0(\mathcal{I})}\left(\Pi +1_{\mathcal{J}} g+G\right)& t>0, \ x_1\in \mathcal{I},
\end{cases}
\end{equation}
\begin{equation}\label{je15:01}
W(0,\cdot\,)=W^{0} \quad \text{in} \ \Omega,\quad \zeta(0,\cdot\,)=\zeta_1^0, \quad \partial_t\zeta (0,\cdot\,)=\zeta_2^0 \quad \text{in} \ \mathcal{I},
\end{equation}
with the control
\[
(f,g)=\mathbb{E}_T([W^0, \zeta_1^0, \zeta_2^0],(F,G))
\]
satisfies
\begin{multline}\label{je12:07}
\left\| W\right\|_{L^2_{\sigma_3}(0,T;H^2(\Omega))\cap C^0_{\sigma_3}([0,T];H^1(\Omega))\cap H^1_{\sigma_3}(0,T;L^2(\Omega))} +
\left\| \Pi\right\|_{L^2_{\sigma_3}(0,T;H^1_0(\Omega))}
\\
+
\left\| \zeta \right\|_{L^2_{\sigma_3}(0,T;H^4(\mathcal{I}))} +
\left\| \zeta \right\|_{C^0_{\sigma_3}([0,T];H^3(\mathcal{I}))} +
\left\| \zeta \right\|_{H^1_{\sigma_3}(0,T;H^2(\mathcal{I}))}
\\
+
\left\| \zeta \right\|_{C^1_{\sigma_3}([0,T];H^1(\mathcal{I}))} +
\left\| \zeta \right\|_{H^2_{\sigma_3}([0,T];L^2(\mathcal{I}))}\\
\lesssim
\left\| [W^0, \zeta_1^0, \zeta_2^0]\right\|_{\mathcal{V}} +
\left\|(F,G)\right\|_{L^2_{\sigma_1}(0,T;L^2(\Omega)\times L^2(\mathcal{I}))}.
\end{multline}
Moreover, we can assume
\begin{equation}\label{je15:05}
\sigma_3^2 \lesssim \sigma_1.
\end{equation}
\end{prop}

We are now in a position to prove \cref{T03} and thus \cref{T02}.
\begin{proof}[Proof of \cref{T03}]
Assume that $[W^0, \zeta_1^0, \zeta_2^0]$ satisfies~\eqref{je15:03} for some $R_0$ and let us assume that
\[
\left\|(F,G)\right\|_{L^2_{\sigma_1}(0,T;L^2(\Omega)\times L^2(\mathcal{I}))}\leq R_0.
\]
Applying \cref{P07}, we deduce the existence of a control $(f,g)\in L^2_{\sigma_2}(0,T;L^2(\omega)\times L^2(\mathcal{J}))$ such that the corresponding solution of~\eqref{je15:00}, \eqref{je15:01} satisfies
\begin{multline}\label{je15:27}
\left\| W\right\|_{L^2_{\sigma_3}(0,T;H^2(\Omega))\cap C^0_{\sigma_3}([0,T];H^1(\Omega))\cap H^1_{\sigma_3}(0,T;L^2(\Omega))} +
\left\| \Pi\right\|_{L^2_{\sigma_3}(0,T;H^1_0(\Omega))}
\\
+
\left\| \zeta \right\|_{L^2_{\sigma_3}(0,T;H^4(\mathcal{I}))} +
\left\| \zeta \right\|_{C^0_{\sigma_3}([0,T];H^3(\mathcal{I}))} +
\left\| \zeta \right\|_{H^1_{\sigma_3}(0,T;H^2(\mathcal{I}))}
\\
+
\left\| \zeta \right\|_{C^1_{\sigma_3}([0,T];H^1(\mathcal{I}))} +
\left\| \zeta \right\|_{H^2_{\sigma_3}([0,T];L^2(\mathcal{I}))}
\leq CR_0
\end{multline}
for some constant $C>0$. Using the Sobolev embeddings, we have in particular that
\begin{equation}\label{je15:29}
\left\| \zeta \right\|_{C^0([0,T];W^{2,\infty}(\mathcal{I}))} \leq CR_0
\end{equation}
for some constant $C>0$. This yields that for $R_0$ small enough, \eqref{me12:28} holds and we can consider the change of variables of \cref{sec_chg}. We thus define $X$, $a$, $\mathbb{F}$ and $\mathbb{G}$ by respectively, \eqref{defX}, \eqref{defa}, \eqref{me22:00} and~\eqref{me22:01}. Moreover, following the arguments in~\cite{MR2745779, MR3619065} and using~\eqref{je15:05}, one can show that
\begin{equation}\label{je16:03}
\left\| \mathbb{F}_\zeta(W,\Pi) \right\|_{L^2_{\sigma_1}(0,T;L^2(\Omega))} +
\left\| \mathbb{G}_\zeta(W) \right\|_{L^2_{\sigma_1}(0,T;L^2(\mathcal{I}))} \leq CR_0^2,
\end{equation}
and in particular for $R_0$ small enough, the closed set
\[
B_{R_0}:=\left\{(F,G)\in L^2_{\sigma_1}(0,T;L^2(\Omega)\times L^2(\mathcal{I})) \ ; \
\left\|(F,G)\right\|_{L^2_{\sigma_1}(0,T;L^2(\Omega)\times L^2(\mathcal{I}))}\leq R_0\right\}
\]
is invariant under the map
\[
\mathcal{Z} : (F,G) \to (\mathbb{F}_\zeta(W,\Pi),\mathbb{G}_\zeta(W)).
\]
One can also show that for $R_0>0$ small enough, the above map is a strict contraction on $B_{R_0}$. Using the Banach fixed point we deduce the existence of fixed point $(F,G)$ for $\mathcal{Z}$. One can notice that the corresponding solution $(W,\Pi,\zeta)$ of~\eqref{je15:00}--\eqref{je15:01} verifies the conclusion of \cref{T03}.
\end{proof}



\appendix

\section{Technical results}\label{sec_tec}

\subsection{A Carleman estimates for the damped beam equation}\label{sec_tecA}

The proof of \cref{CarBeam} follows directly from the proof done in~\cite{Sourav}. The differences with respect to this article is the weight in time and the powers of $s$, $\mu$, and $\xi_0$. For sake of completeness, we give here a brief sketch of the proof of \cref{CarBeam} by using what is already done in~\cite{Sourav}.

We recall that $\varphi_0$ and $\xi_0$ are given by~\eqref{1131123bis}. We set
\begin{equation}\label{ve16:02}
f_\eta:=\partial_{t}^{2} \eta +\partial_{x_{1}}^{4}\eta-\partial^{2}_{x_1} \eta -\partial_{t}\partial_{x_1}^{2} \eta,
\end{equation}
\begin{equation}\label{ve16:03}
\zeta:=e^{s\varphi_0} \xi_0^r \eta.
\end{equation}

We say that a function $g$ is $\lot$ (lower order term) if it satisfies for some $\varepsilon_1, \varepsilon_2 \geq 0$, $\varepsilon_1\varepsilon_2\neq 0$,
\begin{multline*}
|g| \lesssim s^{-\varepsilon_1}\lambda^{-\varepsilon_2}\xi_0^{-\varepsilon_1} \left(s^{7/2}\mu^4 \xi_0^{7/2}\left|\zeta\right| +s^{5/2}\mu^3 \xi_0^{5/2}\left|\partial_{x_1} \zeta\right| +s^{3/2}\mu^2 \xi_0^{3/2}\left(\left|\partial_{x_1}^2 \zeta\right|+\left|\partial_{t} \zeta\right| \right)
\right. \\ \left. +s^{1/2}\mu \xi_0^{1/2}\left(\left|\partial_{x_1}^3 \zeta\right|+\left|\partial_{t}\partial_{x_1} \zeta\right| \right) +s^{-1/2} \xi_0^{-1/2}\left(\left|\partial_{x_1}^4 \zeta\right|+\left|\partial_{t}\partial_{x_1}^2 \zeta\right| +\left|\partial_{t}^2 \zeta\right|\right)
\right).
\end{multline*}

From the Leibniz formula
\[
e^{s\varphi_0} \xi_0^r \frac{\partial^4}{\partial x_1^4} \left(e^{-s\varphi_0} \xi_0^{-r} \zeta\right) =\sum_{\alpha=0}^4 \binom{4}{\alpha} e^{s\varphi_0} \frac{\partial^{\alpha}}{\partial x_1^{\alpha}} \left(e^{-s\varphi_0} \zeta\right)
\xi_0^r \frac{\partial^{4-\alpha}}{\partial x_1^{4-\alpha}} \left(\xi_0^{-r}\right).
\]
From~\eqref{1700}, we obtain
\[
\left| \xi_0^r \frac{\partial^{4-\alpha}}{\partial x_1^{4-\alpha}} \left(\xi_0^{-r}\right) \right| \lesssim \mu^{4-\alpha}
\]
and thus a direct computation and~\eqref{1700} yield that for $s\geq s_0 (T^2+T^4)$,
\[
\sum_{\alpha=0}^3 \binom{4}{\alpha} e^{s\varphi_0} \frac{\partial^{\alpha}}{\partial x_1^{\alpha}} \left(e^{-s\varphi_0} \zeta\right)
\xi_0^r \frac{\partial^{4-\alpha}}{\partial x_1^{4-\alpha}} \left(\xi_0^{-r}\right) = \lot.
\]
We also deduce from~\eqref{1700} that
\begin{multline*}
e^{s\varphi_0} \frac{\partial^4}{\partial x_1^4} \left(e^{-s\varphi_0} \zeta\right) =-4s^3 \left(\partial_{x_1}\varphi_0\right)^3 \partial_{x_1}\zeta +6 s^2 \left(\partial_{x_1}\varphi_0\right)^2 \partial_{x_1}^2\zeta -4s \partial_{x_1}\varphi_0 \partial_{x_1}^3 \zeta +\partial_{x_1}^4 \zeta +s^4 \left(\partial_{x_1}\varphi_0\right)^4 \zeta
\\
-12 s^3 \left(\partial_{x_1}\varphi_0\right)^2 \partial_{x_1}^2\varphi_0 \zeta +\lot
\end{multline*}
and thus
\begin{multline}\label{ve12:40}
e^{s\varphi_0} \xi_0^r \frac{\partial^4}{\partial x_1^4} \left(e^{-s\varphi_0} \xi_0^{-r} \zeta\right) =-4s^3 \left(\partial_{x_1}\varphi_0\right)^3 \partial_{x_1}\zeta +6 s^2 \left(\partial_{x_1}\varphi_0\right)^2 \partial_{x_1}^2\zeta -4s \partial_{x_1}\varphi_0 \partial_{x_1}^3 \zeta +\partial_{x_1}^4 \zeta
\\
+s^4 \left(\partial_{x_1}\varphi_0\right)^4 \zeta -12 s^3 \left(\partial_{x_1}\varphi_0\right)^2 \partial_{x_1}^2\varphi_0 \zeta +\lot.
\end{multline}
Note that
\[
-12 s^3 \left(\partial_{x_1}\varphi_0\right)^2 \partial_{x_1}^2\varphi_0 \zeta=\lot,
\]
but we follow the trick of~\cite{Sourav} to keep this term in order to show the Carleman estimate.

We can show similarly that
\begin{equation}\label{ve12:45}
e^{s\varphi_0} \xi_0^r \frac{\partial^2}{\partial x_1^2} \left(e^{-s\varphi_0} \xi_0^{-r} \zeta\right) =\lot.
\end{equation}
We also have
\[
e^{s\varphi_0} \xi_0^r \frac{\partial^2}{\partial t^2} \left(e^{-s\varphi_0} \xi_0^{-r} \zeta\right) =e^{s\varphi_0} \frac{\partial^2}{\partial t^2} \left(e^{-s\varphi_0} \zeta\right) +2 e^{s\varphi_0} \frac{\partial}{\partial t} \left(e^{-s\varphi_0} \zeta\right)
\xi_0^r \frac{\partial}{\partial t} \left(\xi_0^{-r}\right) +\zeta \xi_0^r \frac{\partial^{2}}{\partial t^{2}} \left(\xi_0^{-r}\right).
\]
Thus, using~\eqref{1700}, for $s\geq s_0 (T^2+T^4)$,
\begin{equation}\label{ve15:10}
e^{s\varphi_0} \xi_0^r \frac{\partial^2}{\partial t^2} \left(e^{-s\varphi_0} \xi_0^{-r} \zeta\right) =\partial_t^2 \zeta+\lot.
\end{equation}
Finally,
\begin{multline*}
e^{s\varphi_0} \xi_0^r \frac{\partial}{\partial_t}\frac{\partial^2}{\partial x_1^2} \left(e^{-s\varphi_0} \xi_0^{-r} \zeta\right) = e^{s\varphi_0} \frac{\partial^2}{\partial x_1^2} \left(e^{-s\varphi_0} \zeta\right)
\xi_0^r \frac{\partial}{\partial t} \left(\xi_0^{-r}\right) +e^{s\varphi_0} \frac{\partial}{\partial t}\frac{\partial^2}{\partial x_1^2} \left(e^{-s\varphi_0} \zeta\right)
\\
+2e^{s\varphi_0} \frac{\partial}{\partial x_1} \left(e^{-s\varphi_0} \zeta\right)
\xi_0^r \frac{\partial}{\partial t} \frac{\partial}{\partial x_1} \left(\xi_0^{-r}\right) +2e^{s\varphi_0} \frac{\partial}{\partial t}\frac{\partial}{\partial x_1} \left(e^{-s\varphi_0} \zeta\right)
\xi_0^r \frac{\partial}{\partial x_1}\left(\xi_0^{-r}\right)
\\
+\zeta
\xi_0^r \frac{\partial}{\partial t} \frac{\partial^2}{\partial x_1^2} \left(\xi_0^{-r}\right) +e^{s\varphi_0} \frac{\partial}{\partial t}\left(e^{-s\varphi_0} \zeta\right)
\xi_0^r \frac{\partial^2}{\partial x_1^2}\left(\xi_0^{-r}\right).
\end{multline*}
From~\eqref{1700}, for $s\geq s_0 (T^2+T^4)$, and for $p=0,1,2$,
\[
\left| \xi_0^r \frac{\partial^p}{\partial t^p}\frac{\partial^{\alpha}}{\partial x_1^{\alpha}} \left(\xi_0^{-r}\right) \right| \lesssim \mu^{\alpha} \xi_0^{p/2}
\]
and
\[
e^{s\varphi_0} \xi_0^r \frac{\partial}{\partial_t}\frac{\partial^2}{\partial x_1^2} \left(e^{-s\varphi_0} \xi_0^{-r} \zeta\right) = e^{s\varphi_0} \frac{\partial}{\partial t}\frac{\partial^2}{\partial x_1^2} \left(e^{-s\varphi_0} \zeta\right) +\lot.
\]
Consequently, using~\eqref{1700}, we deduce that for $s\geq s_0 (T^2+T^4)$,
\[
e^{s\varphi_0} \xi_0^r \frac{\partial}{\partial t}\frac{\partial^2}{\partial x_1^2} \left(e^{-s\varphi_0} \xi_0^{-r} \zeta\right) = s^2 \left(\partial_{x_1} \varphi_0\right)^2 \partial_t \zeta -2s \partial_{x_1} \varphi_0 \partial_t\partial_{x_1} \zeta +\partial_t\partial_{x_1}^2 \zeta +\lot.
\]
Gathering~\eqref{ve12:40}, \eqref{ve12:45}, \eqref{ve15:10}, and the above relation and combining them with~\eqref{ve16:02} and~\eqref{ve16:03}, we deduce
\begin{equation}\label{ve16:05}
M_1\zeta+M_2\zeta = e^{s\varphi_0} \xi_0^r f_\eta + \lot,
\end{equation}
with
\[
M_1 \zeta:=s^4 \left(\partial_{x_1} \varphi_0\right)^4 \zeta + 6 s^2 \left(\partial_{x_1} \varphi_0\right)^2 \partial_{x_1}^2 \zeta +\partial_{x_1}^4 \zeta+2s \left(\partial_{x_1} \varphi_0\right) \partial_t\partial_{x_1}\zeta+\partial_t^2 \zeta,
\]
and
\[
M_2 \zeta := -4 s^3 \left(\partial_{x_1} \varphi_0\right)^3 \partial_{x_1}\zeta -4s \partial_{x_1} \varphi_0 \partial_{x_1}^3\zeta -\partial_t \partial_{x_1}^2 \zeta -s^2 \left(\partial_{x_1} \varphi_0\right)^2 \partial_{t}\zeta -12 s^3 \left(\partial_{x_1} \varphi_0\right)^2 \partial_{x_1}^2 \varphi_0 \zeta.
\]
In what follows, we say that a term $G$ is a $\LOT$ (Lower Order Term) if there exist $\varepsilon_1\geq 0$, $\varepsilon_2 \geq 0$, $\varepsilon_1\varepsilon_2\neq 0$, such that
\begin{multline}\label{ve16:19}
\left| G \right| \lesssim \iint_{(0,T)\times\mathcal{I}} s^{-\varepsilon_1}\lambda^{-\varepsilon_2}\xi_0^{-\varepsilon_1}
\bigg(s^{7}\mu^{8} \xi_{0}^{7} |\zeta |^{2} + s^{5}\mu^{6} \xi_{0}^{5} | \partial_{x_1}\zeta |^{2} + s^{3}\mu^{4}\xi_{0}^{3} \left(| \partial_{x_1}^{2}\zeta |^{2}+ | \partial_{t}\zeta |^{2}\right)
\\
+s\mu^{2}\xi_{0} \left(| \partial_{x_1}^{3}\zeta |^{2}+ |\partial_{t}\partial_{x_1}\zeta |^{2}\right) +s^{-1}\xi_{0}^{-1} \left(\left| \partial_{x_1}^{4}\zeta \right|^{2}+ | \partial_{t}^2\zeta |^{2}+ | \partial_{t}\partial_{x_1}^2 \zeta |^{2}\right)
\bigg)
\ \dt\, \dx_1
\end{multline}
Then, we deduce
\begin{multline}\label{ve16:17}
\left\| M_1\zeta\right\|_{L^2(0,T;L^2(\Omega))}^2 +\left\| M_2\zeta\right\|_{L^2(0,T;L^2(\Omega))}^2 +2\iint_{(0,T)\times \mathcal{I}} M_1 \zeta \cdot M_2\zeta \ \dx_1 \,\dt
\\
=
\left\| e^{s\varphi_0} \xi_0^r f_\eta \zeta\right\|_{L^2(0,T;L^2(\Omega))}^2 + \LOT.
\end{multline}
Writing $I_{i,j}$ for the product term of the $i$-th term of $M_1\zeta$ with the $j$-th term if $M_2\zeta$, we have
\[
\iint_{(0,T)\times \mathcal{I}} M_1 \zeta \cdot M_2\zeta \ \dx_1 \,\dt=\sum_{i,j\in \{1,\ldots,5\}} I_{i,j}
\]
and we have to estimate all the terms $I_{i,j}$. This is done in a precise way in~\cite{Sourav} using~\eqref{1700}. For instance, by integration by parts,
\begin{multline*}
I_{1,2}=-4s^5 \iint_{(0,T)\times \mathcal{I}} \left(\partial_{x_1} \varphi_0\right)^5 \zeta \partial_{x_1}^3 \zeta \ \dx_1\,\dt =-30 s^5 \iint_{(0,T)\times \mathcal{I}} \left(\partial_{x_1} \varphi_0\right)^4 \partial_{x_1}^2 \varphi_0 \left(\partial_{x_1} \zeta\right)^2 \ \dx_1\,\dt
\\
+20 s^5 \iint_{(0,T)\times \mathcal{I}} \left[4 \left(\partial_{x_1} \varphi_0\right)^3 \left(\partial_{x_1}^2 \varphi_0\right)^2 +\left(\partial_{x_1} \varphi_0\right)^4 \partial_{x_1}^3 \varphi_0
\right]
\zeta \partial_{x_1} \zeta \ \dx_1\,\dt
\end{multline*}
and using~\eqref{1700}, we deduce, as in~\cite{Sourav} that
\[
I_{1,2}=-4s^5 \iint_{(0,T)\times \mathcal{I}} \left(\partial_{x_1} \varphi_0\right)^5 \zeta \partial_{x_1}^3 \zeta \ \dx_1\,\dt =-30 s^5 \iint_{(0,T)\times \mathcal{I}} \left(\partial_{x_1} \varphi_0\right)^4 \partial_{x_1}^2 \varphi_0 \left(\partial_{x_1} \zeta\right)^2 \ \dx_1\,\dt +\LOT.
\]
Then, following the computations in~\cite{Sourav}, we find
\begin{multline*}
\iint_{(0,T)\times \mathcal{I}} M_1 \zeta \cdot M_2\zeta \ \dx_1 \,\dt
\geq c
\iint_{(0,T)\times\mathcal{I}}
\bigg(s^{7}\mu^{8} \xi_{0}^{7} |\zeta |^{2} + s^{5}\mu^{6} \xi_{0}^{5} | \partial_{x_1}\zeta |^{2} + s^{3}\mu^{4}\xi_{0}^{3} \left(| \partial_{x_1}^{2}\zeta |^{2}+ | \partial_{t}\zeta |^{2}\right)
\\
+s\mu^{2}\xi_{0} \left(| \partial_{x_1}^{3}\zeta |^{2}+ |\partial_{t}\partial_{x_1}\zeta |^{2}\right) +s^{-1}\xi_{0}^{-1} \left(\left| \partial_{x_1}^{4}\zeta \right|^{2}+ | \partial_{t}^2\zeta |^{2}+ | \partial_{t}\partial_{x_1}^2 \zeta |^{2}\right)
\bigg)
\ \dt\, \dx_1
\\
-C
\iint_{(0,T)\times\mathcal{J}_0}
\bigg(s^{7}\mu^{8} \xi_{0}^{7} |\zeta |^{2} + s^{5}\mu^{6} \xi_{0}^{5} | \partial_{x_1}\zeta |^{2} + s^{3}\mu^{4}\xi_{0}^{3} \left(| \partial_{x_1}^{2}\zeta |^{2}+ | \partial_{t}\zeta |^{2}\right)
\\
+s\mu^{2}\xi_{0} \left(| \partial_{x_1}^{3}\zeta |^{2}+ |\partial_{t}\partial_{x_1}\zeta |^{2}\right) +s^{-1}\xi_{0}^{-1} \left(\left| \partial_{x_1}^{4}\zeta \right|^{2}+ | \partial_{t}^2\zeta |^{2}+ | \partial_{t}\partial_{x_1}^2 \zeta |^{2}\right)
\bigg)
\ \dt\, \dx_1.
\end{multline*}
Then by using standard techniques as in~\cite{Sourav}, we deduce the result.



\subsection{A Carleman estimate for the heat equation}\label{sec_tecB}
We give here a sketch of the proof of \cref{CarVel}. We recall that $\varphi$ and $\xi$ are defined by~\eqref{1131123} and we define
\begin{equation}\label{1131123-2}
\begin{aligned}
\widetilde\varphi(t,x_1,x_2) &:= \frac{1}{\ell(t)^2}(e^{-\lambda\psi_\Omega(x_1,x_2)+\mu \psi_{\mathcal{I}}(x_1)+8\lambda\Psi}-e^{10\lambda\Psi}),\\
\widetilde\xi(t,x_1,x_2)&:=\frac{1}{\ell(t)^2}e^{-\lambda\psi_\Omega(x_1,x_2)+\mu \psi_{\mathcal{I}}(x_1)+8\lambda\Psi}.
\end{aligned}
\end{equation}
We also recall that $\psi$ is defined by~\eqref{defpsi} and we define
\begin{equation}\label{defpsi-2}
\widetilde\psi(x_{1},x_{2}):=\frac{\mu}{\lambda}\psi_{\mathcal{I}}(x_{1})-\psi_{\Omega}(x_{1},x_{2}).
\end{equation}
We have in particular
\[
\varphi=\frac{1}{\ell^2}(e^{\lambda \left(\psi+8\Psi\right)}-e^{10\lambda\Psi}),\quad
\xi:=\frac{1}{\ell^2}e^{\lambda \left(\psi+8\Psi\right)},
\quad
\widetilde\varphi=\frac{1}{\ell^2}(e^{\lambda \left(\widetilde\psi+8\Psi\right)}-e^{10\lambda\Psi}),\quad
\widetilde\xi:=\frac{1}{\ell^2}e^{\lambda \left(\widetilde\psi+8\Psi\right)}.
\]
We set
\[
v:=e^{s\varphi} \xi^r u, \quad
\widetilde v:=e^{s\widetilde \varphi} \widetilde \xi^r u.
\]
Using~\eqref{rt1055}, we have
\begin{equation}\label{sa15:33}
\psi=\widetilde \psi, \quad \varphi=\widetilde \varphi, \quad \xi=\widetilde \xi, \quad v=\widetilde v, \quad
\frac{\partial \psi}{\partial x_1}=\frac{\partial \widetilde \psi}{\partial x_1}, \quad
\frac{\partial \psi}{\partial n}=-\frac{\partial \widetilde \psi}{\partial n}
\quad \text{on} \ (0,T)\times \partial\Omega,
\end{equation}
and using that $\frac{\partial u_2}{\partial n}=0$ on $(0,T)\times \Gamma_1$, we deduce that
\begin{equation}\label{sa15:34}
\frac{\partial v_2}{\partial n}=-\frac{\partial \widetilde v_2}{\partial n} \quad \text{on} \ (0,T)\times \Gamma_1.
\end{equation}
Since $\mu=\mu_0$, taking $\lambda_0\geq \mu_0$ and $\lambda \geq \lambda_0$, we deduce that
\[
\left| \psi\right| +\left| \nabla \psi\right| + \left| \nabla^2 \psi\right| +
\left|\widetilde \psi\right| +\left|\widetilde \nabla \psi\right| + \left|\widetilde \nabla^2 \psi\right|
\lesssim 1.
\]
There exists $s_0>0$ such that for $s\geq s_0(T^2+T^4)$,
\[
1\leq s\xi,
\;\; \left| \nabla^{\alpha} \xi \right|+\left| \nabla^{\alpha} \varphi \right| \lesssim \lambda^{\alpha} \xi \;\; (k\geq 1),
\;\; \left|\partial_t \nabla^{\alpha} \xi \right|+\left|\partial_t \nabla^{\alpha} \varphi \right| \lesssim \lambda^{\alpha} T\xi^{3/2},
\;\; \left|\partial_t^2 \nabla^{\alpha} \xi \right|+\left|\partial_t^2 \nabla^{\alpha} \varphi \right| \lesssim \lambda^{\alpha} T^2 \xi^{2},
\]
\[
1\leq s\widetilde\xi,
\;\; \left| \nabla^{\alpha} \widetilde\xi \right|+\left| \nabla^{\alpha} \widetilde\varphi \right| \lesssim \lambda^{\alpha} \widetilde\xi \;\; (k\geq 1),
\;\; \left|\partial_t \nabla^{\alpha} \widetilde\xi \right|+\left|\partial_t \nabla^{\alpha} \widetilde\varphi \right| \lesssim \lambda^{\alpha} T\widetilde\xi^{3/2},
\;\; \left|\partial_t^2 \nabla^{\alpha} \widetilde\xi \right|+\left|\partial_t^2 \nabla^{\alpha} \widetilde\varphi \right| \lesssim \lambda^{\alpha} T^2 \widetilde\xi^{2},
\]
Using the above relations and following standard calculations (see, for instance, \cite{FCGBGP}), we obtain the existence of $s_0, c, C, \widetilde c, \widetilde C>0$ such that for $s\geq s_0(T^2+T^4)$,
\begin{multline}\label{1332203}
c\iint_{(0,T)\times \Omega} \left(s^3\lambda^4\xi^3 \left|v\right|^2 +s\lambda^2\xi \left|\nabla v\right|^2+\frac{1}{s\xi} \left|\Delta v\right|^2 +\frac{1}{s\xi} \left|\partial_t v\right|^2 \right) \ \dx\dt
\\
-\iint_{(0,T)\times \partial \Omega} 2s^3\lambda^3\xi^3 \left|\nabla \psi \right|^2 \frac{\partial \psi}{\partial n} \left| v\right|^2 \ \dx_1\dt -\iint_{(0,T)\times \partial \Omega} 4s\lambda^2\xi \left|\nabla \psi \right|^2 \frac{\partial v}{\partial n} \cdot v \ \dx_1\dt
\\
-\iint_{(0,T)\times \partial \Omega} 4s\lambda\xi \left(\nabla v \nabla \psi\right) \cdot \frac{\partial v}{\partial n} \ \dx_1\dt +\iint_{(0,T)\times \partial \Omega} 2s\lambda\xi \frac{\partial \psi}{\partial n} \left|\nabla v\right|^2\ \dx_1\dt
\\
-\iint_{(0,T)\times \partial \Omega} 2\partial_t v \cdot \frac{\partial v}{\partial n} \ \dx_1\dt -\iint_{(0,T)\times \partial \Omega} 2s^2\lambda\xi \frac{\partial \psi}{\partial n} \partial_t \varphi \left| v\right|^2\ \dx_1\dt
\\
\leq C\left(\iint_{(0,T)\times \Omega} \left|\partial_t u -\Delta u\right|^2 \xi^{2r} e^{2s\varphi} \ \dx\dt +\iint_{(0,T)\times \omega_1} s^3\lambda^4\xi^3 \left|v\right|^2 \ \dx\dt\right)
\end{multline}
and
\begin{multline}\label{1332204}
\widetilde c\iint_{(0,T)\times \Omega} \left(s^3\lambda^4\widetilde \xi^3 \left| \widetilde v \right|^2 +s\lambda^2\widetilde \xi \left|\nabla \widetilde v\right|^2+\frac{1}{s\widetilde \xi} \left|\Delta \widetilde v\right|^2 +\frac{1}{s\widetilde \xi} \left|\partial_t \widetilde v\right|^2 \right) \ \dx\dt
\\
-\iint_{(0,T)\times \partial \Omega} 2s^3\lambda^3\widetilde \xi^3 \left|\nabla \widetilde \psi \right|^2 \frac{\partial \widetilde \psi}{\partial n} \left| \widetilde v\right|^2 \ \dx_1\dt -\iint_{(0,T)\times \partial \Omega} 4s\lambda^2\widetilde \xi \left|\nabla \widetilde \psi \right|^2 \frac{\partial \widetilde v}{\partial n} \cdot \widetilde v \ \dx_1\dt
\\
-\iint_{(0,T)\times \partial \Omega} 4s\lambda\widetilde \xi \left(\nabla \widetilde v \nabla \widetilde \psi\right) \cdot \frac{\partial \widetilde v}{\partial n} \ \dx_1\dt +\iint_{(0,T)\times \partial \Omega} 2s\lambda\widetilde \xi \frac{\partial \widetilde \psi}{\partial n} \left|\nabla \widetilde v\right|^2\ \dx_1\dt
\\
-\iint_{(0,T)\times \partial \Omega} 2\partial_t \widetilde v \cdot \frac{\partial \widetilde v}{\partial n} \ \dx_1\dt -\iint_{(0,T)\times \partial \Omega} 2s^2\lambda\widetilde \xi \frac{\partial \widetilde \psi}{\partial n} \partial_t \widetilde \varphi \left| \widetilde v\right|^2\ \dx_1\dt
\\
\leq \widetilde C\left(\iint_{(0,T)\times \Omega} \left| \partial_t u -\Delta u\right|^2 \widetilde \xi^{2r} e^{2s\widetilde \varphi} \ \dx\dt +\iint_{(0,T)\times \omega_1} s^3\lambda^4\widetilde \xi^3 \left| \widetilde v\right|^2 \ \dx\dt\right).
\end{multline}

Summing~\eqref{1332203} and~\eqref{1332204} and using~\eqref{sa15:34}, \eqref{sa15:34}, we deduce that
\begin{multline}
c\iint_{(0,T)\times \Omega} \left(s^3\lambda^4\xi^3 \left| v\right|^2 +s\lambda^2\xi \left|\nabla v\right|^2+\frac{1}{s\xi} \left|\Delta v\right|^2 +\frac{1}{s\xi} \left|\partial_t v\right|^2 \right) \ \dx\dt
\\
+\widetilde c\iint_{(0,T)\times \Omega} \left(s^3\lambda^4\widetilde \xi^3 \left| \widetilde v\right|^2 +s\lambda^2\widetilde \xi \left|\nabla \widetilde v\right|^2+\frac{1}{s\widetilde \xi} \left|\Delta \widetilde v\right|^2 +\frac{1}{s\widetilde \xi} \left|\partial_t \widetilde v\right|^2 \right) \ \dx\dt
\\
\leq C\left(\iint_{(0,T)\times \Omega} \left|\partial_t u -\Delta u\right|^2 \xi^{2r} e^{2s\varphi} \ \dx\dt +\iint_{(0,T)\times \omega_0} s^3\lambda^4\xi^3 \left|v\right|^2 \ \dx\dt\right)
\\
+\widetilde C\left(\iint_{(0,T)\times \Omega} \left|\partial_t u -\Delta u\right|^2 \widetilde \xi^{2r} e^{2s\widetilde \varphi} \ \dx\dt +\iint_{(0,T)\times \omega_1} s^3\lambda^4\widetilde \xi^3 \left|\widetilde v\right|^2 \ \dx\dt\right).
\end{multline}
Then, using that $\widetilde \varphi \leq \varphi$ and $\widetilde \xi \leq \xi$, we deduce that
\begin{multline}
\iint_{(0,T)\times \Omega} \left(s^3\lambda^4\xi^3 \left| v\right| ^2 +s\lambda^2\xi \left|\nabla v\right|^2+\frac{1}{s\xi} \left|\Delta v\right|^2 +\frac{1}{s\xi} \left|\partial_t v\right|^2 \right) \ \dx\dt
\\
\lesssim
\iint_{(0,T)\times \Omega} \left|\partial_t u -\Delta u\right|^2 \xi^{2r} e^{2s\varphi} \ \dx\dt +\iint_{(0,T)\times \omega_1} s^3\lambda^4\xi^3 \left|v\right|^2 \ \dx\dt.
\end{multline}
From the elliptical regularity of the Laplace operator, we deduce
\begin{multline}
\iint_{(0,T)\times \Omega} \left(s^3\lambda^4\xi^3 \left| v\right| ^2+s\lambda^2\xi \left|\nabla v\right|^2+\frac{1}{s\xi} \left|\nabla^2 v\right|^2 +\frac{1}{s\xi} \left|\partial_t v\right|^2 \right) \ \dx\dt
\\
\lesssim
\iint_{(0,T)\times \Omega} \left|\partial_t u -\Delta u\right|^2 \xi^{2r} e^{2s\varphi} \ \dx\dt +\iint_{(0,T)\times \omega_1} s^3\lambda^4\xi^3 \left|v\right|^2 \ \dx\dt
\end{multline}
and with standard computations, we can come back to $u$:
\begin{multline}
\iint_{(0,T)\times \Omega} e^{2s\varphi} \left(s^{2r+3}\lambda^{2r+4}\xi^{2r+3} \left| u\right| ^2+ s^{2r+1}\lambda^{2r+2}\xi^{2r+1} \left|\nabla u\right|^2
\right. \\ \left. +s^{2r-1}\lambda^{2r}\xi^{2r-1} \left|\nabla^2 u\right|^2 +s^{2r-1}\lambda^{2r}\xi^{2r-1} \left|\partial_t u\right|^2 \right) \ \dx\dt
\\
\lesssim
\iint_{(0,T)\times \Omega} \left| \partial_t u -\Delta u\right|^2 (s\lambda \xi)^{2r} e^{2s\varphi} \ \dx\dt +\iint_{(0,T)\times \omega_1} s^{2r+3}\lambda^{2r+4}\xi^{2r+3} e^{2s\varphi} \left| u\right|^2 \ \dx\dt.
\end{multline}
We deduce \cref{CarVel} by taking $r=1/2$.



%\section*{Acknowledgements} The second author was partially supported by the Agence Nationale de la Recherche, Project TRECOS, ANR-20-CE40-0009.



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